doi = {10.11588/heidok.00005743},
url = {https://doi.org/10.11588/heidok.00005743}
}
+
+@article{CiarletRiavart1972interpolation,
+ author = {P. Ciarlet and P. Raviart},
+ title = {General Langrange and Hermite interpolation in $\mathbb{R}^{n}$ with applications to finite element methods},
+ journal = {Archive for rational mechanics and analysis},
+ volume = {46},
+ pages = {177-199},
+ year = {1972}
+}
--- /dev/null
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2021 - 2022 by the deal.II authors
+//
+// This file is part of the deal.II library.
+//
+// The deal.II library is free software; you can use it, redistribute
+// it, and/or modify it under the terms of the GNU Lesser General
+// Public License as published by the Free Software Foundation; either
+// version 2.1 of the License, or (at your option) any later version.
+// The full text of the license can be found in the file LICENSE.md at
+// the top level directory of deal.II.
+//
+// ---------------------------------------------------------------------
+
+
+
+#ifndef dealii_hermite_polynomials
+#define dealii_hermite_polynomials
+
+#include <deal.II/base/config.h>
+
+#include <deal.II/base/exceptions.h>
+#include <deal.II/base/point.h>
+#include <deal.II/base/polynomial.h>
+#include <deal.II/base/subscriptor.h>
+
+#include <deal.II/lac/full_matrix.h>
+
+#include <algorithm>
+#include <exception>
+#include <memory>
+#include <vector>
+
+
+
+DEAL_II_NAMESPACE_OPEN
+
+/**
+ * @addtogroup Polynomials
+ * @{
+ */
+
+namespace Polynomials
+{
+ /**
+ * This class implements Hermite interpolation polynomials (see
+ * @cite CiarletRiavart1972interpolation) enforcing the maximum
+ * posible level of regularity $r$ in the FEM basis given a
+ * polynomial degree of $2r+1$. The polynomials all represent
+ * either a non-zero shape value or derivative at $y=0$ and $y=1$
+ * on the reference interval $y \in [0,1]$.
+ *
+ * Indices $j = 0, 1, \dots, r$ refer to polynomials corresponding
+ * to a non-zero derivative (or shape value for $j=0$) of
+ * order $j$ at $x=0$, and indices $j = r+1, r+2, \dots, 2r+1$
+ * refer to polynomials with a non-zero derivative of order
+ * $j-(r+1)$ (or value for $j=r+1$) at $x=1$. The basis is rescaled
+ * so that a function corresponding to a non-zero $j^{th}$
+ * derivative has derivative value $j! 4^{j}$ at the corresponding
+ * node.
+ */
+ class HermitePoly : public Polynomial<double>
+ {
+ public:
+ /**
+ * Constructor for an individual Hermite polynomial. We write f_{j}
+ * for a polynomial that has a non-zero $j^{th}$ derivative at $y=0$
+ * and $g_{j}$ for a polynomial with a non-zero $j^{th}$ derivative
+ * at $y=1$, meaning $f_{j}$ will have @p index $=j$ and $g_{j}$ will
+ * have @p index $= j + \mathtt{regularity} + 1$. The resulting
+ * polynomials will be degree $2\times \mathtt{regularity} +1$
+ * and obey the following conditions:
+ *
+ * @f{align*}{
+ * &\begin{matrix}
+ * \left. \frac{d^{i}}{dy^{i}} f_{j}(y) \right\vert_{y=0}
+ * = i! 4^{i} \delta_{i, j}, \hfill
+ * &\qquad \hfill 0 \leq i \leq \mathtt{regularity}, \\
+ * \left. \frac{d^{i}}{dy^{i}} f_{j}(y) \right\vert_{y=1}
+ * = 0, \hfill &\qquad \hfill 0 \leq i \leq \mathtt{regularity},
+ * \end{matrix} \qquad 0 \leq j \leq \mathtt{regularity}, \\
+ * &\begin{matrix}
+ * \left. \frac{d^{i}}{dy^{i}} g_{j}(y) \right\vert_{y=0}
+ * = 0, \hfill &\qquad \hfill 0 \leq i \leq \mathtt{regularity}, \\
+ * \left. \frac{d^{i}}{dy^{i}} g_{j}(y) \right\vert_{y=1}
+ * = i! 4^{i} \delta_{i, j}, \hfill
+ * &\qquad \hfill 0 \leq i \leq \mathtt{regularity},
+ * \end{matrix} \qquad 0 \leq j \leq \mathtt{regularity},
+ * @f}
+ *
+ * where $\delta_{i,j}$ is equal to $1$ whenever $i=j$,
+ * and equal to $0$ otherwise. These polynomials have explicit
+ * formulas given by
+ *
+ * @f{align*}{
+ * f_{j}(y) &= 4^{j} y^{j} (1-y)^{\mathtt{regularity}+1}
+ * \sum_{k=0}^{\mathtt{regularity} - j} \;^{\mathtt{regularity} + k} C_{k}
+ * y^{k}, \\ g_{j}(y) &= 4^{j} y^{\mathtt{regularity}+1} (y-1)^{j}
+ * \sum_{k=0}^{\mathtt{regularity} - j} \;^{\mathtt{regularity} + k} C_{k}
+ * (1-y)^{k},
+ * @f}
+ *
+ * where $^{n} C_{r} = \frac{n!}{r!(n-r)!}$ is the $r^{th}$ binomial
+ * coefficient of degree $n, \; 0 \leq r \leq n$.
+ *
+ * @param regularity The highest derivative for which the basis
+ * is used to enforce regularity.
+ * @param index The local index of the generated polynomial in the
+ * Hermite basis.
+ */
+ HermitePoly(const unsigned int regularity, const unsigned int index);
+
+ /**
+ * This function generates a vector of Polynomial objects
+ * representing a complete basis of degree $2\times\mathtt{regularity} +1$
+ * on the reference interval $[0,1]$.
+ *
+ * @param regularity The generated basis can be used to strongly
+ * enforce continuity in all derivatives up to and including this
+ * order.
+ */
+ static std::vector<Polynomial<double>>
+ generate_complete_basis(const unsigned int regularity);
+
+ protected:
+ /**
+ * Degree of the polynomial basis being used.
+ */
+ unsigned int degree;
+
+ /**
+ * The order of the highest derivative in which the Hermite
+ * basis can be used to impose continuity across element
+ * boundaries. It's related to the degree $p$ by
+ * $p = 2 \times\mathtt{regularity} +1$.
+ */
+ unsigned int regularity;
+
+ /**
+ * This variable stores the derivative that the shape function
+ * corresponds to at the element boundary given by <code>side</code>.
+ */
+ unsigned int side_index;
+
+ /**
+ * This stores whether the shape function corresponds to a non-zero
+ * value or derivative at $x=0$ on the reference interval
+ * ($\mathtt{side} =0$), or at $x=1$ ($\mathtt{side} =1$).
+ */
+ unsigned int side;
+ };
+} // namespace Polynomials
+/** @} */
+
+DEAL_II_NAMESPACE_CLOSE
+
+#endif
--- /dev/null
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2021 - 2022 by the deal.II authors
+//
+// This file is part of the deal.II library.
+//
+// The deal.II library is free software; you can use it, redistribute
+// it, and/or modify it under the terms of the GNU Lesser General
+// Public License as published by the Free Software Foundation; either
+// version 2.1 of the License, or (at your option) any later version.
+// The full text of the license can be found in the file LICENSE.md at
+// the top level directory of deal.II.
+//
+// ---------------------------------------------------------------------
+
+#ifndef dealii_fe_hermite
+#define dealii_fe_hermite
+
+#include <deal.II/base/config.h>
+#include <deal.II/base/polynomials_hermite.h>
+
+#include <deal.II/fe/fe_poly.h>
+
+#include <string>
+#include <vector>
+
+DEAL_II_NAMESPACE_OPEN
+
+/**
+ * @addtogroup fe
+ * @{
+ */
+
+
+
+/**
+ * This class implements a Hermite interpolation basis of maximum regularity elements
+ * (see @cite CiarletRiavart1972interpolation). These are always of odd polynomial
+ * degree, have regularity $r=\frac{p-1}{2}$ and are defined up to polynomial degree $p=13$.
+ *
+ * Each node has $(r+1)^{d}$ degrees of freedom (DoFs) assigned to it,
+ * corresponding to the different combinations of directional derivatives up to
+ * the required regularity at that node. DoFs at each node are not consecutive
+ * in either the global or local ordering, due to the tensor product
+ * construction of the basis. The ordering is determined by the direction of
+ * the derivative each function corresponds to; first by $x$-derivatives, then
+ * $y$, then $z$. Locally over each element the DoFs are ordered similarly. See
+ * below for the local ordering for $r=1$, where DoFs are ordered
+ * from 0 to $(2r+2)^{\mathtt{dim} }-1$:
+ *
+ * <code>FE_Hermite<1>(1)</code>
+ *
+ * @verbatim
+ * (0)________________(2)
+ * (1) (3)
+ * @endverbatim
+ *
+ * <code>FE_Hermite<2>(1)</code>:
+ *
+ * @verbatim
+ * ( 8, 9)__________(10,11)
+ * (12,13) (14,15)
+ * | |
+ * | |
+ * | |
+ * | |
+ * | |
+ * | |
+ * ( 0, 1)__________( 2, 3)
+ * ( 4, 5) ( 6, 7)
+ * @endverbatim
+ *
+ * <code>FE_Hermite<3>(1)</code>:
+ *
+ * @verbatim
+ * (40,41,44,45)__(42,43,46,47) (40,41,44,45)__(42,43,46,47)
+ * (56,57,60,61) (58,59,62,63) (56,57,60,61) (58,59,62,63)
+ * / | | / / |
+ * / | | / / |
+ * / | | / / |
+ *(32,33,36,37)| | (32,33,36,37)__(34,35,38,39)
+ *(48,49,52,53)| | (48,49,52,53) (50,51,54,55)
+ * | | | | | |
+ * |( 8,9,12,13 )__(10,11,14,15) | |(10,11,14,15)
+ * |(24,25,28,29) (26,27,30,31) | |(26,27,30,31)
+ * | / / | | /
+ * | / / | | /
+ * | / / | | /
+ * ( 0,1,4,5 )___( 2,3,6,7 ) ( 0,1,4,5 )___( 2,3,6,7 )
+ * (16,17,20,21) (18,19,22,23) (16,17,20,21) (18,19,22,23)
+ * @endverbatim
+ *
+ * Note that while the number of functions defined on each cell appears large,
+ * due to the increased regularity constraints many of these functions are
+ * shared between elements.
+ */
+template <int dim, int spacedim = dim>
+class FE_Hermite : public FE_Poly<dim, spacedim>
+{
+public:
+ /**
+ * Constructor that creates an Hermite finite element that imposes
+ * continuity in derivatives up to and including order @p regularity.
+ */
+ FE_Hermite<dim, spacedim>(const unsigned int regularity);
+
+ // Other functions
+ /**
+ * Returns <code>FE_Hermite<dim,spacedim>(regularity)</code> as a
+ * <code>std::string</code> with @p dim, @p spacedim and @p regularity
+ * replaced with the correct values.
+ */
+ virtual std::string
+ get_name() const override;
+
+ /**
+ * @copydoc dealii::FiniteElement::clone()
+ */
+ virtual std::unique_ptr<FiniteElement<dim, spacedim>>
+ clone() const override;
+
+ /**
+ * Returns the mapping between lexicographic and hierarchic numbering
+ * schemes for Hermite. See the class documentation for diagrams of
+ * examples of lexicographic numbering for Hermite elements.
+ */
+ std::vector<unsigned int>
+ get_lexicographic_to_hierarchic_numbering() const;
+
+ /**
+ * This re-implements FiniteElement::fill_fe_values() for a Hermite
+ * polynomial basis, to account for the more complicated shape function
+ * re-scaling that a Hermite basis requires. Since the idea of a Hermite
+ * basis is to strongly impose derivative continuities at the element
+ * boundaries, it is necessary to account for any changes to derivatives
+ * from those on the reference element due to the current cell mapping.
+ *
+ * At present this is done by only allowing certain cell mappings
+ * (currently only MappingCartesian) that guarantee rectangular cells
+ * which ensures that the directions of derivatives does not change.
+ * The derivative continuity can then be enforced by re-scaling shape
+ * functions so that the magnitude of each derivative is equivalent
+ * to the derivative on the reference interval.
+ */
+ virtual void
+ fill_fe_values(
+ const typename Triangulation<dim, spacedim>::cell_iterator &cell,
+ const CellSimilarity::Similarity cell_similarity,
+ const Quadrature<dim> & quadrature,
+ const Mapping<dim, spacedim> & mapping,
+ const typename Mapping<dim, spacedim>::InternalDataBase &mapping_internal,
+ const dealii::internal::FEValuesImplementation::MappingRelatedData<dim,
+ spacedim>
+ & mapping_data,
+ const typename FiniteElement<dim, spacedim>::InternalDataBase &fe_internal,
+ dealii::internal::FEValuesImplementation::FiniteElementRelatedData<dim,
+ spacedim>
+ &output_data) const override;
+
+ using FiniteElement<dim, spacedim>::fill_fe_face_values;
+
+ /**
+ * This re-implements FiniteElement::fill_fe_face_values() for Hermite
+ * polynomial bases, for the same reasons as described for
+ * FEHermite::fill_fe_values().
+ */
+ virtual void
+ fill_fe_face_values(
+ const typename Triangulation<dim, spacedim>::cell_iterator &cell,
+ const unsigned int face_no,
+ const hp::QCollection<dim - 1> & quadrature,
+ const Mapping<dim, spacedim> & mapping,
+ const typename Mapping<dim, spacedim>::InternalDataBase &mapping_internal,
+ const dealii::internal::FEValuesImplementation::MappingRelatedData<dim,
+ spacedim>
+ & mapping_data,
+ const typename FiniteElement<dim, spacedim>::InternalDataBase &fe_internal,
+ dealii::internal::FEValuesImplementation::FiniteElementRelatedData<dim,
+ spacedim>
+ &output_data) const override;
+
+
+
+protected:
+ /**
+ * Wrapper function for FE_Poly::get_data() that ensures relevant
+ * shape value data is copied to the new data object as well.
+ */
+ virtual std::unique_ptr<
+ typename FiniteElement<dim, spacedim>::InternalDataBase>
+ get_data(
+ const UpdateFlags update_flags,
+ const Mapping<dim, spacedim> &mapping,
+ const Quadrature<dim> & quadrature,
+ dealii::internal::FEValuesImplementation::FiniteElementRelatedData<dim,
+ spacedim>
+ &output_data) const override
+ {
+ std::unique_ptr<typename FiniteElement<dim, spacedim>::InternalDataBase>
+ data_ptr = FE_Poly<dim, spacedim>::get_data(update_flags,
+ mapping,
+ quadrature,
+ output_data);
+ auto &data =
+ dynamic_cast<typename FE_Poly<dim, spacedim>::InternalData &>(*data_ptr);
+
+ const unsigned int n_q_points = quadrature.size();
+
+ if ((update_flags & update_values) &&
+ ((output_data.shape_values.n_rows() > 0) &&
+ (output_data.shape_values.n_cols() == n_q_points)))
+ data.shape_values = output_data.shape_values;
+
+ return data_ptr;
+ }
+
+
+
+private:
+ /**
+ * Variable storing the order of the highest derivative that the current
+ * @p FE_Hermite object can enforce continuity for. Here the order of
+ * derivative only counts in one spatial direction, so the derivative
+ * $\frac{d^{2}f}{dx \; dy}$ would be counted as a first order derivative
+ * of $f$, as an example.
+ */
+ unsigned int regularity;
+
+
+
+public:
+ /**
+ * Returns the regularity of the Hermite FE basis.
+ */
+ inline unsigned int
+ get_regularity() const
+ {
+ return this->regularity;
+ };
+};
+
+/** @} */
+
+
+
+DEAL_II_NAMESPACE_CLOSE
+
+#endif
#include <deal.II/fe/fe.h>
#include <deal.II/fe/fe_dgp.h>
#include <deal.II/fe/fe_dgq.h>
+#include <deal.II/fe/fe_hermite.h>
#include <deal.II/fe/fe_poly.h>
#include <deal.II/fe/fe_pyramid_p.h>
#include <deal.II/fe/fe_q.h>
return false;
// then check if the base element is supported or not
- if (dynamic_cast<const FE_Poly<dim, spacedim> *>(fe_ptr) != nullptr)
+ if (dynamic_cast<const FE_Poly<dim, spacedim> *>(fe_ptr) != nullptr &&
+ dynamic_cast<const FE_Hermite<dim, spacedim> *>(fe_ptr) ==
+ nullptr)
{
const FE_Poly<dim, spacedim> *fe_poly_ptr =
dynamic_cast<const FE_Poly<dim, spacedim> *>(fe_ptr);
#include <deal.II/dofs/dof_tools.h>
#include <deal.II/fe/fe.h>
+#include <deal.II/fe/fe_hermite.h>
#include <deal.II/fe/fe_values.h>
#include <deal.II/fe/mapping_q1.h>
const unsigned int n_components = fe.n_components();
const unsigned int n_function_components =
boundary_functions.begin()->second->n_components;
- const bool fe_is_system = (n_components != 1);
+ const bool fe_is_system = (n_components != 1);
+ const bool fe_is_hermite =
+ (dynamic_cast<const FE_Hermite<dim, spacedim> *>(&fe) != nullptr);
const bool fe_is_primitive = fe.is_primitive();
copy_data.cell = cell;
UpdateFlags update_flags =
UpdateFlags(update_values | update_JxW_values | update_normal_vectors |
update_quadrature_points);
+ if (fe_is_hermite)
+ update_flags = update_flags | update_mapping;
FEFaceValues<dim, spacedim> fe_values(mapping, fe, q, update_flags);
// two variables for the coefficient, one for the two cases
const FiniteElement<dim, spacedim> &fe = dof.get_fe();
const unsigned int n_components = fe.n_components();
+ const bool fe_is_hermite =
+ (dynamic_cast<const FE_Hermite<dim, spacedim> *>(&fe) != nullptr);
- Assert(matrix.n() == dof.n_boundary_dofs(boundary_functions),
+ Assert(fe_is_hermite ||
+ matrix.n() == dof.n_boundary_dofs(boundary_functions),
ExcInternalError());
+ (void)fe_is_hermite;
+
Assert(matrix.n() == matrix.m(), ExcInternalError());
Assert(matrix.n() == rhs_vector.size(), ExcInternalError());
Assert(boundary_functions.size() != 0, ExcInternalError());
#include <deal.II/numerics/vector_tools_common.h>
#include <deal.II/numerics/vector_tools_constraints.h>
#include <deal.II/numerics/vector_tools_evaluate.h>
+#include <deal.II/numerics/vector_tools_hermite.h>
#include <deal.II/numerics/vector_tools_integrate_difference.h>
#include <deal.II/numerics/vector_tools_interpolate.h>
#include <deal.II/numerics/vector_tools_mean_value.h>
#include <deal.II/numerics/vector_tools_boundary.templates.h>
#include <deal.II/numerics/vector_tools_constraints.templates.h>
+#include <deal.II/numerics/vector_tools_hermite.templates.h>
#include <deal.II/numerics/vector_tools_integrate_difference.templates.h>
#include <deal.II/numerics/vector_tools_interpolate.templates.h>
#include <deal.II/numerics/vector_tools_mean_value.templates.h>
--- /dev/null
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2021 - 2022 by the deal.II authors
+//
+// This file is part of the deal.II library.
+//
+// The deal.II library is free software; you can use it, redistribute
+// it, and/or modify it under the terms of the GNU Lesser General
+// Public License as published by the Free Software Foundation; either
+// version 2.1 of the License, or (at your option) any later version.
+// The full text of the license can be found in the file LICENSE.md at
+// the top level directory of deal.II.
+//
+// ---------------------------------------------------------------------
+
+#ifndef vector_tools_hermite_h
+#define vector_tools_hermite_h
+
+#include <deal.II/base/config.h>
+
+#include <deal.II/base/function.h>
+#include <deal.II/base/quadrature.h>
+#include <deal.II/base/quadrature_lib.h>
+#include <deal.II/base/types.h>
+
+#include <deal.II/dofs/dof_handler.h>
+
+#include <deal.II/fe/mapping.h>
+
+#include <map>
+#include <vector>
+
+DEAL_II_NAMESPACE_OPEN
+
+namespace VectorTools
+{
+ /**
+ * @name Domain and boundary projection for Hermite elements
+ * @{
+ */
+
+ /**
+ * @p Enumeration representing the type of boundary condition to be enforced.
+ * Since Hermite finite elements can strongly impose continuity in various
+ * derivatives as well as value at element boundaries, they can also be used
+ * to impose boundary conditions such as Neumann conditions directly to reduce
+ * the size of the computational system.
+ */
+ enum HermiteBoundaryType
+ {
+ hermite_dirichlet,
+ hermite_neumann,
+ hermite_2nd_derivative,
+ };
+
+
+
+ /**
+ * Enforces boundary conditions by projecting onto the Hermite finite element
+ * space at the boundary.
+ *
+ * This is required even in the case of Dirichlet boundary conditions where
+ * similar functions already exist for other bases, because Hermite assigns
+ * degrees of freedom directly onto element boundaries that have a zero shape
+ * value across entire element faces. For most existing methods this can
+ * create a singular mass matrix unless special steps are taken to remove
+ * those degrees of freedom from the boundary system.
+ *
+ * Note that only Dirichlet, Neumann and second degree Neumann
+ * conditions are currently supported.
+ */
+ template <int dim, int spacedim = dim, typename Number = double>
+ void
+ project_hermite_boundary_values(
+ const Mapping<dim, spacedim> & mapping_h,
+ const DoFHandler<dim, spacedim> &dof_handler,
+ const std::map<types::boundary_id, const Function<spacedim, Number> *>
+ & boundary_functions,
+ const Quadrature<dim - 1> & quadrature,
+ const HermiteBoundaryType projection_mode,
+ std::map<types::global_dof_index, Number> &boundary_values,
+ std::vector<unsigned int> component_mapping = {});
+
+
+
+ /**
+ * The following function is a specialisation of the above for the case where
+ * Dirichlet boundary conditions should be enforced.
+ */
+ template <int dim, int spacedim = dim, typename Number = double>
+ void
+ project_hermite_boundary_values(
+ const Mapping<dim, spacedim> & mapping_h,
+ const DoFHandler<dim, spacedim> &dof_handler,
+ const std::map<types::boundary_id, const Function<spacedim, Number> *>
+ & boundary_functions,
+ const Quadrature<dim - 1> & quadrature,
+ std::map<types::global_dof_index, Number> &boundary_values,
+ std::vector<unsigned int> component_mapping = {});
+
+
+
+ /**
+ * Projection function for use with Hermite finite elements. Usually the
+ * existing VectorTools::project() function will work with Hermite, however
+ * setting any of the flags for enforcing boundary conditions to true would
+ * cause errors, either by setting all derivatives to 0 at the boundary or
+ * creating a singular boundary mass matrix. Since the Hermite mass matrix can
+ * become poorly conditioned for higher polynomial degrees it is often useful
+ * to perform separate boundary and domain projection, which means a separate
+ * projection function for Hermite elements is necessary for their effective
+ * use.
+ */
+ template <int dim, typename VectorType, int spacedim>
+ void
+ project_hermite(
+ const Mapping<dim, spacedim> & mapping,
+ const DoFHandler<dim, spacedim> & dofhandler,
+ const AffineConstraints<typename VectorType::value_type> & constraints,
+ const Quadrature<dim> & quadrature,
+ const Function<spacedim, typename VectorType::value_type> &function,
+ VectorType & vec,
+ const bool enforce_zero_boundary = false,
+ const Quadrature<dim - 1> &q_boundary = (dim > 1 ? QGauss<dim - 1>(2) :
+ Quadrature<dim - 1>(0)),
+ const bool project_to_boundary_first = false);
+
+ /** @} */
+} // namespace VectorTools
+
+DEAL_II_NAMESPACE_CLOSE
+
+#endif
--- /dev/null
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2021 - 2022 by the deal.II authors
+//
+// This file is part of the deal.II library.
+//
+// The deal.II library is free software; you can use it, redistribute
+// it, and/or modify it under the terms of the GNU Lesser General
+// Public License as published by the Free Software Foundation; either
+// version 2.1 of the License, or (at your option) any later version.
+// The full text of the license can be found in the file LICENSE.md at
+// the top level directory of deal.II.
+//
+// ---------------------------------------------------------------------
+
+#ifndef vector_tools_hermite_template_h
+#define vector_tools_hermite_template_h
+
+#include <deal.II/base/quadrature.h>
+#include <deal.II/base/quadrature_lib.h>
+#include <deal.II/base/table.h>
+#include <deal.II/base/template_constraints.h>
+#include <deal.II/base/utilities.h>
+
+#include <deal.II/dofs/dof_tools.h>
+
+#include <deal.II/fe/fe_face.h>
+#include <deal.II/fe/fe_hermite.h>
+#include <deal.II/fe/fe_tools.h>
+#include <deal.II/fe/fe_values.h>
+#include <deal.II/fe/mapping_cartesian.h>
+
+#include <deal.II/lac/dynamic_sparsity_pattern.h>
+#include <deal.II/lac/precondition.h>
+#include <deal.II/lac/solver_cg.h>
+#include <deal.II/lac/solver_control.h>
+#include <deal.II/lac/sparse_direct.h>
+#include <deal.II/lac/sparse_matrix.h>
+#include <deal.II/lac/sparsity_pattern.h>
+#include <deal.II/lac/sparsity_tools.h>
+#include <deal.II/lac/vector.h>
+
+#include <deal.II/numerics/matrix_tools.h>
+#include <deal.II/numerics/vector_tools_boundary.h>
+#include <deal.II/numerics/vector_tools_project.h>
+#include <deal.II/numerics/vector_tools_rhs.h>
+#include <deal.II/numerics/vector_tools_hermite.h>
+
+
+DEAL_II_NAMESPACE_OPEN
+
+
+
+/*
+ * Define MappingHermite as an alias for MappingCartesian. In the future it
+ * may be possible to relax the requirements for a mapping class used with
+ * Hermite, which is why this name is reserved here.
+ */
+
+template <int dim, int spacedim = dim>
+using MappingHermite = MappingCartesian<dim, spacedim>;
+
+namespace VectorTools
+{
+ // Internal namespace for implement Hermite boundary projection methods
+ namespace internal
+ {
+ template <int dim, int spacedim = dim, typename Number = double>
+ void
+ get_constrained_hermite_boundary_dofs(
+ const DoFHandler<dim, spacedim> &dof_handler,
+ const std::map<types::boundary_id, const Function<spacedim, Number> *>
+ & boundary_functions,
+ const unsigned int position,
+ std::vector<types::global_dof_index> &dof_to_boundary,
+ types::global_dof_index & no_constrained_dofs)
+ {
+ AssertDimension(dof_handler.n_dofs(), dof_to_boundary.size());
+
+ std::fill_n(dof_to_boundary.begin(),
+ dof_handler.n_dofs(),
+ numbers::invalid_dof_index);
+
+ /*
+ * Create a look-up table for finding constrained dofs on all
+ * 4 or six faces of reference cell
+ */
+ const unsigned int degree = dof_handler.get_fe().degree;
+ const unsigned int regularity =
+ dynamic_cast<const FE_Hermite<dim> &>(dof_handler.get_fe())
+ .get_regularity();
+ const unsigned int dofs_per_face = dof_handler.get_fe().n_dofs_per_face();
+ const unsigned int dofs_per_cell = Utilities::pow( degree + 1, dim);
+ const unsigned int constrained_dofs_per_face =
+ dofs_per_face / (regularity + 1);
+
+ AssertDimension(dofs_per_face,
+ (regularity + 1) * Utilities::pow(degree + 1, dim - 1));
+
+ std::vector<types::global_dof_index> dofs_on_cell(dofs_per_cell);
+ Table<2, double> constrained_to_local_indices(2 * dim,
+ constrained_dofs_per_face);
+
+ /*
+ * Use knowledge of the local degree numbering for this version,
+ * saving expensive calls to reinit
+ */
+ const std::vector<unsigned int> l2h =
+ dynamic_cast<const FE_Hermite<dim> &>(dof_handler.get_fe())
+ .get_lexicographic_to_hierarchic_numbering();
+ // const auto test_cell = dof_handler.begin();
+
+ for (unsigned int d = 0, batch_size = 1; d < dim;
+ ++d, batch_size *= degree + 1)
+ for (unsigned int i = 0; i < constrained_dofs_per_face; ++i)
+ {
+ const unsigned int local_index = i % batch_size;
+ const unsigned int batch_index = i / batch_size;
+
+ unsigned int index =
+ local_index +
+ (batch_index * (degree + 1) + position) * batch_size;
+ unsigned int correction =
+ batch_size * (regularity + 1);
+ Assert(index + correction < dofs_per_cell,
+ ExcDimensionMismatch(index + correction, dofs_per_cell));
+
+ // test_cell->face(2 * d)->get_dof_indices(
+ // dofs_on_face, test_cell->active_fe_index());
+ constrained_to_local_indices(2 * d, i) = l2h[index];
+
+ // test_cell->face(2 * d + 1)->get_dof_indices(
+ // dofs_on_face, test_cell->active_fe_index());
+ constrained_to_local_indices(2 * d + 1, i) = l2h[index + correction];
+ }
+
+ std::set<types::boundary_id> selected_boundary_components;
+
+ for (auto i = boundary_functions.cbegin(); i != boundary_functions.cend();
+ ++i)
+ selected_boundary_components.insert(i->first);
+
+ Assert(selected_boundary_components.find(
+ numbers::internal_face_boundary_id) ==
+ selected_boundary_components.end(),
+ DoFTools::ExcInvalidBoundaryIndicator());
+
+ no_constrained_dofs = 0;
+
+ for (const auto &cell : dof_handler.active_cell_iterators())
+ for (const unsigned int f : cell->face_indices())
+ {
+ AssertDimension(cell->get_fe().n_dofs_per_face(f),
+ dofs_per_face);
+
+ // Check if face is on selected boundary section
+ if (selected_boundary_components.find(
+ cell->face(f)->boundary_id()) !=
+ selected_boundary_components.end())
+ {
+ cell->get_dof_indices(dofs_on_cell); //, cell->active_fe_index());
+
+ for (unsigned int i = 0; i < constrained_dofs_per_face; ++i)
+ {
+ const types::global_dof_index index =
+ dofs_on_cell[constrained_to_local_indices(f, i)];
+ Assert(index < dof_to_boundary.size(),
+ ExcDimensionMismatch(index, dof_to_boundary.size()));
+
+ if (dof_to_boundary[index] == numbers::invalid_dof_index)
+ dof_to_boundary[index] = no_constrained_dofs++;
+ }
+ }
+ }
+
+ Assert((no_constrained_dofs !=
+ dof_handler.n_boundary_dofs(boundary_functions)) ||
+ (regularity == 0),
+ ExcInternalError());
+ }
+
+
+
+ template <int dim, int spacedim = dim, typename Number = double>
+ void
+ do_hermite_direct_projection(
+ const Mapping<dim, spacedim> & mapping_h,
+ const DoFHandler<dim, spacedim> &dof_handler,
+ const std::map<types::boundary_id, const Function<spacedim, Number> *>
+ & boundary_functions,
+ const Quadrature<dim - 1> & quadrature,
+ const unsigned int position,
+ std::map<types::global_dof_index, Number> &boundary_values,
+ std::vector<unsigned int> component_mapping = {})
+ {
+ // Return immediately if no constraint functions are provided
+ if (boundary_functions.size() == 0)
+ return;
+
+ // Check the mapping object is of a compatible mapping class
+ Assert((dynamic_cast<const MappingHermite<dim, spacedim> *>(&mapping_h) !=
+ nullptr) ||
+ (dynamic_cast<const MappingCartesian<dim> *>(&mapping_h) !=
+ nullptr),
+ ExcInternalError());
+
+ // For dim=1, the problem simplifies to interpolation at the boundaries
+ if (dim == 1)
+ {
+ Assert(component_mapping.size() == 0, ExcNotImplemented());
+
+ for (const auto &cell : dof_handler.active_cell_iterators())
+ for (const unsigned int direction :
+ GeometryInfo<dim>::face_indices())
+ if (cell->at_boundary(direction) &&
+ (boundary_functions.find(
+ cell->face(direction)->boundary_id()) !=
+ boundary_functions.end()))
+ {
+ const Function<spacedim, Number> ¤t_function =
+ *boundary_functions
+ .find(cell->face(direction)->boundary_id())
+ ->second;
+
+ const FiniteElement<dim, spacedim> &fe_herm =
+ dof_handler.get_fe();
+
+ AssertDimension(fe_herm.n_components(),
+ current_function.n_components);
+
+ Vector<Number> boundary_value(fe_herm.n_components());
+
+ if (boundary_value.size() == 1)
+ boundary_value(0) =
+ current_function.value(cell->vertex(direction));
+ else
+ current_function.vector_value(cell->vertex(direction),
+ boundary_value);
+
+ boundary_values[cell->vertex_dof_index(
+ direction, position, cell->active_fe_index())] =
+ boundary_value(
+ fe_herm.face_system_to_component_index(position).first);
+ }
+
+ return;
+ }
+
+ // dim=2 or higher, needs actual projection
+ if (component_mapping.size() == 0)
+ {
+ AssertDimension(dof_handler.get_fe().n_components(),
+ boundary_functions.begin()->second->n_components);
+
+ component_mapping.resize(dof_handler.get_fe().n_components());
+
+ for (unsigned int i = 0; i < component_mapping.size(); ++i)
+ component_mapping[i] = i;
+ }
+ else
+ AssertDimension(component_mapping.size(),
+ dof_handler.get_fe().n_components());
+
+ std::vector<types::global_dof_index> dof_to_boundary_mapping(
+ dof_handler.n_dofs(), numbers::invalid_dof_index);
+ types::global_dof_index next_boundary_index;
+
+ get_constrained_hermite_boundary_dofs(dof_handler,
+ boundary_functions,
+ position,
+ dof_to_boundary_mapping,
+ next_boundary_index);
+
+ //Check to see if the boundaries are homogeneous
+ bool zero_indicator = true;
+ for (const auto& func_pair : boundary_functions)
+ {
+ zero_indicator = zero_indicator & (dynamic_cast<const Functions::ZeroFunction<dim>*>(func_pair.second) != nullptr);
+ if (!zero_indicator) break;
+ }
+
+ if (zero_indicator)
+ {
+ for (unsigned int i = 0; i < dof_to_boundary_mapping.size(); ++i)
+ if (dof_to_boundary_mapping[i] != numbers::invalid_dof_index)
+ boundary_values[i] = 0;
+
+ return;
+ }
+
+ SparsityPattern sparsity;
+
+ {
+ DynamicSparsityPattern dsp(next_boundary_index, next_boundary_index);
+ DoFTools::make_boundary_sparsity_pattern(dof_handler,
+ boundary_functions,
+ dof_to_boundary_mapping,
+ dsp);
+
+ sparsity.copy_from(dsp);
+ }
+
+ // Assert mesh is not partially refined
+ // TODO: Add functionality on partially refined meshes
+ int level = -1;
+
+ for (const auto &cell : dof_handler.active_cell_iterators())
+ for (auto f : cell->face_indices())
+ if (cell->at_boundary(f))
+ {
+ if (level == -1)
+ level = cell->level();
+ else
+ Assert(level == cell->level(),
+ ExcMessage("The mesh you use in projecting "
+ "boundary values has hanging nodes"
+ " at the boundary. This would require "
+ "dealing with hanging node constraints"
+ " when solving the linear system on "
+ "the boundary, but this is not "
+ "currently implemented."));
+ }
+
+ // make mass matrix and right hand side
+ SparseMatrix<Number> mass_matrix(sparsity);
+ Vector<Number> rhs(sparsity.n_rows());
+ Vector<Number> boundary_projection(rhs.size());
+
+ MatrixCreator::create_boundary_mass_matrix(
+ mapping_h,
+ dof_handler,
+ quadrature,
+ mass_matrix,
+ boundary_functions,
+ rhs,
+ dof_to_boundary_mapping,
+ static_cast<const Function<spacedim, Number> *>(nullptr),
+ component_mapping);
+
+ {
+ /*
+ * Current implementation uses a direct solver to better account for
+ * the mass matrix being potentially sufficiently ill-conditioned to
+ * prevent iterations from converging.
+ * TODO: Find a good preconditioner to allow more efficient iterative
+ * methods
+ */
+ SparseDirectUMFPACK mass_inv;
+ mass_inv.initialize(mass_matrix);
+ mass_inv.vmult(boundary_projection, rhs);
+ }
+
+ // fill in boundary values
+ for (unsigned int i = 0; i < dof_to_boundary_mapping.size(); ++i)
+ if (dof_to_boundary_mapping[i] != numbers::invalid_dof_index)
+ {
+ AssertIsFinite(boundary_projection(dof_to_boundary_mapping[i]));
+
+ /*
+ * This is where the boundary solution, which only considers the
+ * DOFs that are actually constrained, is applied to the boundary
+ * DOFs in the global solution. Remember: i is the global dof
+ * number, dof_to_boundary_mapping[i] is the number on the boundary
+ * and thus in the solution vector.
+ */
+ boundary_values[i] =
+ boundary_projection(dof_to_boundary_mapping[i]);
+ }
+ return;
+ }
+ } // namespace internal
+
+
+
+ template <int dim, int spacedim, typename Number>
+ void
+ project_hermite_boundary_values(
+ const Mapping<dim, spacedim> & mapping_h,
+ const DoFHandler<dim, spacedim> &dof_handler,
+ const std::map<types::boundary_id, const Function<spacedim, Number> *>
+ & boundary_functions,
+ const Quadrature<dim - 1> & quadrature,
+ const HermiteBoundaryType projection_mode,
+ std::map<types::global_dof_index, Number> &boundary_values,
+ std::vector<unsigned int> component_mapping)
+ {
+ /*
+ * This version implements projected values directly,
+ * so it's necessary to check that this is possible
+ */
+ const bool check_mode =
+ (projection_mode == HermiteBoundaryType::hermite_dirichlet) ||
+ (projection_mode == HermiteBoundaryType::hermite_neumann) ||
+ (projection_mode == HermiteBoundaryType::hermite_2nd_derivative);
+
+ Assert(check_mode, ExcNotImplemented());
+ Assert((dynamic_cast<const MappingHermite<dim, spacedim> *>(&mapping_h) !=
+ nullptr) ||
+ (dynamic_cast<const MappingCartesian<dim> *>(&mapping_h) !=
+ nullptr),
+ ExcMessage("project_hermite_boundary_values should only be"
+ " used with a mapping compatible with Hermite "
+ "finite element methods."));
+ Assert((dynamic_cast<const FE_Hermite<dim> *>(&dof_handler.get_fe()) !=
+ nullptr),
+ ExcMessage("project_hermite_boundary_values should only"
+ " be used with a DOF handler linked to a "
+ "FE_Hermite object."));
+
+ (void)check_mode;
+
+ unsigned int position = 0;
+
+ switch (projection_mode)
+ {
+ case HermiteBoundaryType::hermite_dirichlet:
+ position = 0;
+ break;
+ case HermiteBoundaryType::hermite_neumann:
+ Assert(dof_handler.get_fe().degree > 2,
+ ExcDimensionMismatch(dof_handler.get_fe().degree, 3));
+
+ position = 1;
+ break;
+ case HermiteBoundaryType::hermite_2nd_derivative:
+ Assert(dof_handler.get_fe().degree > 4,
+ ExcDimensionMismatch(dof_handler.get_fe().degree, 5));
+
+ position = 2;
+ break;
+ default:
+ Assert(false, ExcInternalError());
+ }
+
+ internal::do_hermite_direct_projection(mapping_h,
+ dof_handler,
+ boundary_functions,
+ quadrature,
+ position,
+ boundary_values,
+ component_mapping);
+ }
+
+
+
+ template <int dim, int spacedim, typename Number>
+ void
+ project_hermite_boundary_values(
+ const Mapping<dim, spacedim> & mapping_h,
+ const DoFHandler<dim, spacedim> &dof_handler,
+ const std::map<types::boundary_id, const Function<spacedim, Number> *>
+ & boundary_functions,
+ const Quadrature<dim - 1> & quadrature,
+ std::map<types::global_dof_index, Number> &boundary_values,
+ std::vector<unsigned int> component_mapping)
+ {
+ Assert((dynamic_cast<const MappingHermite<dim, spacedim> *>(&mapping_h) !=
+ nullptr) ||
+ (dynamic_cast<const MappingCartesian<dim> *>(&mapping_h) !=
+ nullptr),
+ ExcMessage("project_hermite_boundary_values should only be"
+ " used with a mapping compatible with Hermite "
+ "finite element methods."));
+
+ internal::do_hermite_direct_projection(mapping_h,
+ dof_handler,
+ boundary_functions,
+ quadrature,
+ 0,
+ boundary_values,
+ component_mapping);
+ }
+
+
+
+ template <int dim, typename VectorType, int spacedim>
+ void
+ project_hermite(
+ const Mapping<dim, spacedim> & mapping_h,
+ const DoFHandler<dim, spacedim> & dof,
+ const AffineConstraints<typename VectorType::value_type> & constraints,
+ const Quadrature<dim> & quadrature,
+ const Function<spacedim, typename VectorType::value_type> &function,
+ VectorType & vec,
+ const bool enforce_zero_boundary,
+ const Quadrature<dim - 1> &q_boundary,
+ const bool project_to_boundary_first)
+ {
+ using number = typename VectorType::value_type;
+
+ Assert((dynamic_cast<const MappingHermite<dim, spacedim> *>(&mapping_h) !=
+ nullptr) ||
+ (dynamic_cast<const MappingCartesian<dim> *>(&mapping_h) !=
+ nullptr),
+ ExcMessage(
+ "The function project_hermite() can only be used with a mapping "
+ "compatible with Hermite finite element methods."));
+
+ Assert((dynamic_cast<const FE_Hermite<dim, spacedim> *>(&dof.get_fe()) !=
+ nullptr),
+ ExcMessage("The project_hermite function should only be used"
+ " with a DoF handler associated with an FE_Hermite"
+ " object with the same dim and spacedim as the "
+ "mapping."));
+ Assert(dof.get_fe(0).n_components() == function.n_components,
+ ExcDimensionMismatch(dof.get_fe(0).n_components(),
+ function.n_components));
+ Assert(vec.size() == dof.n_dofs(),
+ ExcDimensionMismatch(vec.size(), dof.n_dofs()));
+
+ // make up boundary values
+ std::map<types::global_dof_index, number> boundary_values;
+ const std::vector<types::boundary_id> active_boundary_ids =
+ dof.get_triangulation().get_boundary_ids();
+
+ if (enforce_zero_boundary)
+ {
+ std::map<types::boundary_id, const Function<spacedim, number> *>
+ boundary;
+
+ for (const auto it : active_boundary_ids)
+ boundary.emplace(std::make_pair(it, nullptr));
+
+ const std::map<types::boundary_id, const Function<spacedim, number> *>
+ new_boundary =
+ static_cast<const std::map<types::boundary_id,
+ const Function<spacedim, number> *>>(
+ boundary);
+
+ std::vector<types::global_dof_index> dof_to_boundary(
+ dof.n_dofs(), numbers::invalid_dof_index);
+ types::global_dof_index end_boundary_dof = 0;
+
+ internal::get_constrained_hermite_boundary_dofs(
+ dof, new_boundary, 0, dof_to_boundary, end_boundary_dof);
+
+ if (end_boundary_dof != 0)
+ for (types::global_dof_index i = 0; i < dof.n_dofs(); ++i)
+ if (dof_to_boundary[i] != numbers::invalid_dof_index)
+ boundary_values.emplace(std::make_pair(i, 0));
+ }
+ else if (project_to_boundary_first)
+ {
+ std::map<types::boundary_id, const Function<spacedim, number> *>
+ boundary_function;
+
+ for (const auto it : active_boundary_ids)
+ boundary_function.emplace(std::make_pair(it, &function));
+
+ const std::map<types::boundary_id, const Function<spacedim, number> *>
+ new_boundary_2 =
+ static_cast<const std::map<types::boundary_id,
+ const Function<spacedim, number> *>>(
+ boundary_function);
+
+ project_hermite_boundary_values<dim, spacedim, number>(
+ mapping_h,
+ dof,
+ new_boundary_2,
+ q_boundary,
+ HermiteBoundaryType::hermite_dirichlet,
+ boundary_values);
+ }
+
+ // check if constraints are compatible (see below)
+ bool constraints_are_compatible = true;
+
+ for (const auto &value : boundary_values)
+ if (constraints.is_constrained(value.first))
+ if ((constraints.get_constraint_entries(value.first)->size() > 0) &&
+ (constraints.get_inhomogeneity(value.first) != value.second))
+ constraints_are_compatible = false;
+
+ // set up mass matrix and right hand side
+ Vector<number> vec_result(dof.n_dofs());
+ SparsityPattern sparsity;
+
+ {
+ DynamicSparsityPattern dsp(dof.n_dofs(), dof.n_dofs());
+ DoFTools::make_sparsity_pattern(dof,
+ dsp,
+ constraints,
+ !constraints_are_compatible);
+
+ sparsity.copy_from(dsp);
+ }
+
+ SparseMatrix<number> mass_matrix(sparsity);
+ Vector<number> tmp(mass_matrix.n());
+
+ /*
+ * If the constraints object does not conflict with the given boundary
+ * values (i.e., it either does not contain boundary values or it contains
+ * the same as boundary_values), we can let it call
+ * distribute_local_to_global straight away, otherwise we need to first
+ * interpolate the boundary values and then condense the matrix and vector
+ */
+ if (constraints_are_compatible)
+ {
+ const Function<spacedim, number> *dummy = nullptr;
+
+ MatrixCreator::create_mass_matrix(mapping_h,
+ dof,
+ quadrature,
+ mass_matrix,
+ function,
+ tmp,
+ dummy,
+ constraints);
+
+ if (boundary_values.size() > 0)
+ MatrixTools::apply_boundary_values(
+ boundary_values, mass_matrix, vec_result, tmp, true);
+ }
+ else
+ {
+ // create mass matrix and rhs at once, which is faster.
+ MatrixCreator::create_mass_matrix(
+ mapping_h, dof, quadrature, mass_matrix, function, tmp);
+ MatrixTools::apply_boundary_values(
+ boundary_values, mass_matrix, vec_result, tmp, true);
+
+ constraints.condense(mass_matrix, tmp);
+ }
+
+ /*
+ * Current implementation uses a direct solver to better deal
+ * with potentially ill-conditioned mass matrices.
+ * TODO: Find a good preconditioner for the Hermite mass matrix
+ */
+ SparseDirectUMFPACK mass_inv;
+ mass_inv.initialize(mass_matrix);
+ mass_inv.vmult(vec_result, tmp);
+
+ constraints.distribute(vec_result);
+
+ // copy vec_result into vec. we can't use vec itself above, since
+ // it may be of another type than Vector<double> and that wouldn't
+ // necessarily go together with the matrix and other functions
+ for (unsigned int i = 0; i < vec.size(); ++i)
+ ::dealii::internal::ElementAccess<VectorType>::set(vec_result(i), i, vec);
+ }
+} // namespace VectorTools
+
+
+
+DEAL_II_NAMESPACE_CLOSE
+
+#endif
polynomials_bernstein.cc
polynomials_bdm.cc
polynomials_bernardi_raugel.cc
+ polynomials_hermite.cc
polynomials_nedelec.cc
polynomials_integrated_legendre_sz.cc
polynomial_space.cc
--- /dev/null
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2021 - 2022 by the deal.II authors
+//
+// This file is part of the deal.II library.
+//
+// The deal.II library is free software; you can use it, redistribute
+// it, and/or modify it under the terms of the GNU Lesser General
+// Public License as published by the Free Software Foundation; either
+// version 2.1 of the License, or (at your option) any later version.
+// The full text of the license can be found in the file LICENSE.md at
+// the top level directory of deal.II.
+//
+// ---------------------------------------------------------------------
+
+#include <deal.II/base/polynomials_hermite.h>
+
+DEAL_II_NAMESPACE_OPEN
+
+namespace Polynomials
+{
+ namespace
+ {
+ std::vector<double>
+ hermite_poly_coeffs(const unsigned int regularity, const unsigned int index)
+ {
+ AssertIndexRange(index, 2 * regularity + 2);
+
+ const unsigned int curr_index = index % (regularity + 1);
+ const bool side = (index > regularity);
+
+ // Protection needed here against underflow errors
+ const int comparison_1 = static_cast<int>(regularity + 1 - curr_index);
+ const int comparison_2 = side ? static_cast<int>(curr_index + 1) :
+ static_cast<int>(regularity + 2);
+
+ std::vector<double> poly_coeffs(2 * regularity + 2, 0.0);
+
+ if (side) // g polynomials
+ {
+ int binomial_1 = (curr_index % 2) ? -1 : 1;
+
+ for (int i = 0; i < comparison_2; ++i)
+ {
+ int inv_binomial = 1;
+
+ for (int j = 0; j < comparison_1; ++j)
+ {
+ int binomial_2 = 1;
+
+ for (int k = 0; k < j + 1; ++k)
+ {
+ poly_coeffs[regularity + i + k + 1] +=
+ binomial_1 * inv_binomial * binomial_2;
+ binomial_2 *= k - j;
+ binomial_2 /= k + 1;
+ }
+ inv_binomial *= regularity + j + 1;
+ inv_binomial /= j + 1;
+ }
+ // Protection needed here against underflow errors
+ binomial_1 *= -static_cast<int>(curr_index - i);
+ binomial_1 /= i + 1;
+ }
+ }
+ else // f polynomials
+ {
+ int binomial = 1;
+
+ for (int i = 0; i < comparison_2; ++i)
+ {
+ int inv_binomial = 1;
+
+ for (int j = 0; j < comparison_1; ++j)
+ {
+ poly_coeffs[curr_index + i + j] += binomial * inv_binomial;
+ inv_binomial *= regularity + j + 1;
+ inv_binomial /= j + 1;
+ }
+ // Protection needed here against underflow errors
+ binomial *= -static_cast<int>(regularity + 1 - i);
+ binomial /= i + 1;
+ }
+ }
+
+ // rescale coefficients by a factor of 4^curr_index to account for reduced
+ // L2-norms
+ double precond_factor = Utilities::pow(4, curr_index);
+ for (auto &it : poly_coeffs)
+ it *= precond_factor;
+
+ return poly_coeffs;
+ }
+ } // namespace
+
+
+
+ HermitePoly::HermitePoly(const unsigned int regularity,
+ const unsigned int index)
+ : Polynomial<double>(hermite_poly_coeffs(regularity, index))
+ , degree(2 * regularity + 1)
+ , regularity(regularity)
+ , side_index(index % (regularity + 1))
+ , side((index > regularity) ? 1 : 0)
+ {}
+
+
+
+ std::vector<Polynomial<double>>
+ HermitePoly::generate_complete_basis(const unsigned int regularity)
+ {
+ std::vector<Polynomial<double>> polys;
+ const unsigned int sz = 2 * regularity + 2;
+
+ for (unsigned int i = 0; i < sz; ++i)
+ polys.push_back(HermitePoly(regularity, i));
+
+ return polys;
+ }
+} // namespace Polynomials
+
+DEAL_II_NAMESPACE_CLOSE
#include <deal.II/dofs/dof_tools.h>
#include <deal.II/fe/fe.h>
+#include <deal.II/fe/fe_hermite.h>
#include <deal.II/fe/fe_tools.h>
#include <deal.II/fe/fe_values.h>
Assert(boundary_ids.find(numbers::internal_face_boundary_id) ==
boundary_ids.end(),
(typename DoFHandler<dim, spacedim>::ExcInvalidBoundaryIndicator()));
- Assert(sparsity.n_rows() == dof.n_boundary_dofs(boundary_ids),
+
+ const bool fe_is_hermite = (dynamic_cast<const FE_Hermite<dim, spacedim> *>(
+ &(dof.get_fe())) != nullptr);
+
+ Assert(fe_is_hermite ||
+ sparsity.n_rows() == dof.n_boundary_dofs(boundary_ids),
ExcDimensionMismatch(sparsity.n_rows(),
dof.n_boundary_dofs(boundary_ids)));
- Assert(sparsity.n_cols() == dof.n_boundary_dofs(boundary_ids),
+ Assert(fe_is_hermite ||
+ sparsity.n_cols() == dof.n_boundary_dofs(boundary_ids),
ExcDimensionMismatch(sparsity.n_cols(),
dof.n_boundary_dofs(boundary_ids)));
+ (void)fe_is_hermite;
+
#ifdef DEBUG
if (sparsity.n_rows() != 0)
{
cell->active_fe_index());
// make sparsity pattern for this cell
+ /*
+ * This loop has been edited to check for invalid dof indices, which
+ * the current implementation of Hermite elements uses to denote
+ * degrees of freedom assigned on the boundary but that have a zero
+ * shape value across the entire boundary.
+ */
for (unsigned int i = 0; i < dofs_per_face; ++i)
for (unsigned int j = 0; j < dofs_per_face; ++j)
- sparsity.add(dof_to_boundary_mapping[dofs_on_this_face[i]],
- dof_to_boundary_mapping[dofs_on_this_face[j]]);
+ if (dof_to_boundary_mapping[dofs_on_this_face[i]] !=
+ numbers::invalid_dof_index &&
+ dof_to_boundary_mapping[dofs_on_this_face[j]] !=
+ numbers::invalid_dof_index)
+ sparsity.add(dof_to_boundary_mapping[dofs_on_this_face[i]],
+ dof_to_boundary_mapping[dofs_on_this_face[j]]);
}
}
fe_dg_vector.cc
fe_enriched.cc
fe_face.cc
+ fe_hermite.cc
fe_nedelec.cc
fe_nedelec_sz.cc
fe_nothing.cc
fe_dg_vector.inst.in
fe_face.inst.in
fe.inst.in
+ fe_hermite.inst.in
fe_nedelec.inst.in
fe_nedelec_sz.inst.in
fe_nothing.inst.in
--- /dev/null
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2021 - 2022 by the deal.II authors
+//
+// This file is part of the deal.II library.
+//
+// The deal.II library is free software; you can use it, redistribute
+// it, and/or modify it under the terms of the GNU Lesser General
+// Public License as published by the Free Software Foundation; either
+// version 2.1 of the License, or (at your option) any later version.
+// The full text of the license can be found in the file LICENSE.md at
+// the top level directory of deal.II.
+//
+// ---------------------------------------------------------------------
+
+#include <deal.II/base/quadrature.h>
+#include <deal.II/base/quadrature_lib.h>
+#include <deal.II/base/table.h>
+#include <deal.II/base/template_constraints.h>
+
+#include <deal.II/dofs/dof_tools.h>
+
+#include <deal.II/fe/fe_face.h>
+#include <deal.II/fe/fe_hermite.h>
+#include <deal.II/fe/fe_tools.h>
+#include <deal.II/fe/fe_values.h>
+#include <deal.II/fe/mapping_cartesian.h>
+
+#include <deal.II/lac/dynamic_sparsity_pattern.h>
+#include <deal.II/lac/precondition.h>
+#include <deal.II/lac/solver_cg.h>
+#include <deal.II/lac/solver_control.h>
+#include <deal.II/lac/sparse_direct.h>
+#include <deal.II/lac/sparse_matrix.h>
+#include <deal.II/lac/sparsity_pattern.h>
+#include <deal.II/lac/sparsity_tools.h>
+#include <deal.II/lac/vector.h>
+
+#include <deal.II/numerics/matrix_tools.h>
+
+#include <algorithm>
+#include <cmath>
+#include <iostream>
+#include <iterator>
+#include <memory>
+#include <sstream>
+
+DEAL_II_NAMESPACE_OPEN
+
+
+
+/*
+ * In the future it may be possible to relax the restrictions on mappings
+ * that can be used with FE_Hermite and implement these in a mapping
+ * class, so the name MappingHermite is used in some places. At the moment
+ * this has not been investigated properly, so MappingHermite is defined
+ * here as an alias for MappingCartesian.
+ */
+template <int dim, int spacedim = dim>
+using MappingHermite = MappingCartesian<dim>;
+
+
+
+namespace internal
+{
+ inline std::vector<unsigned int>
+ get_hermite_dpo_vector(const unsigned int dim, const unsigned int regularity)
+ {
+ std::vector<unsigned int> result(dim + 1, 0);
+ result[0] = Utilities::pow(regularity + 1, dim);
+
+ return result;
+ }
+
+
+
+ /*
+ * Renumbering function. Function needs different levels of for loop nesting
+ * for different values of dim, so different definitions are used for
+ * simplicity.
+ */
+ template <int dim>
+ void
+ hermite_hierarchic_to_lexicographic_numbering(const unsigned int regularity,
+ std::vector<unsigned int> &h2l);
+
+
+
+ template <>
+ void
+ hermite_hierarchic_to_lexicographic_numbering<1>(
+ const unsigned int regularity,
+ std::vector<unsigned int> &h2l)
+ {
+ const unsigned int node_dofs_1d = regularity + 1;
+
+ AssertDimension(h2l.size(), 2 * node_dofs_1d);
+
+ // Assign DOFs at vertices
+ for (unsigned int di = 0; di < 2; ++di)
+ for (unsigned int i = 0; i < node_dofs_1d; ++i)
+ h2l[i + di * node_dofs_1d] = i + di * node_dofs_1d;
+ }
+
+
+
+ template <>
+ void
+ hermite_hierarchic_to_lexicographic_numbering<2>(
+ const unsigned int regularity,
+ std::vector<unsigned int> &h2l)
+ {
+ const unsigned int node_dofs_1d = regularity + 1;
+ const unsigned int dim_dofs_1d = 2 * node_dofs_1d;
+ unsigned int offset = 0;
+
+ AssertDimension(h2l.size(), dim_dofs_1d * dim_dofs_1d);
+
+ // Assign DOFs at vertices
+ for (unsigned int di = 0; di < 2; ++di)
+ for (unsigned int dj = 0; dj < 2; ++dj)
+ {
+ for (unsigned int i = 0; i < node_dofs_1d; ++i)
+ for (unsigned int j = 0; j < node_dofs_1d; ++j)
+ h2l[j + i * node_dofs_1d + offset] =
+ j + i * dim_dofs_1d + (dj + di * dim_dofs_1d) * node_dofs_1d;
+
+ offset += node_dofs_1d * node_dofs_1d;
+ }
+ }
+
+
+
+ template <>
+ void
+ hermite_hierarchic_to_lexicographic_numbering<3>(
+ const unsigned int regularity,
+ std::vector<unsigned int> &h2l)
+ {
+ const unsigned int node_dofs_1d = regularity + 1;
+ const unsigned int node_dofs_2d = node_dofs_1d * node_dofs_1d;
+
+ const unsigned int dim_dofs_1d = 2 * node_dofs_1d;
+ const unsigned int dim_dofs_2d = dim_dofs_1d * dim_dofs_1d;
+
+ unsigned int offset = 0;
+
+ AssertDimension(h2l.size(), dim_dofs_2d * dim_dofs_1d);
+
+ // Assign DOFs at nodes
+ for (unsigned int di = 0; di < 2; ++di)
+ for (unsigned int dj = 0; dj < 2; ++dj)
+ for (unsigned int dk = 0; dk < 2; ++dk)
+ {
+ for (unsigned int i = 0; i < node_dofs_1d; ++i)
+ for (unsigned int j = 0; j < node_dofs_1d; ++j)
+ for (unsigned int k = 0; k < node_dofs_1d; ++k)
+ h2l[k + j * node_dofs_1d + i * node_dofs_2d + offset] =
+ k + j * dim_dofs_1d + i * dim_dofs_2d +
+ node_dofs_1d * (dk + dj * dim_dofs_1d + di * dim_dofs_2d);
+
+ offset += node_dofs_1d * node_dofs_2d;
+ }
+ }
+
+
+
+ template <int dim>
+ inline std::vector<unsigned int>
+ hermite_hierarchic_to_lexicographic_numbering(const unsigned int regularity)
+ {
+ const std::vector<unsigned int> dpo =
+ get_hermite_dpo_vector(dim, regularity);
+ const dealii::FiniteElementData<dim> face_data(dpo, 1, 2 * regularity + 1);
+ std::vector<unsigned int> renumbering(face_data.dofs_per_cell);
+
+ hermite_hierarchic_to_lexicographic_numbering<dim>(regularity, renumbering);
+
+ return renumbering;
+ }
+
+
+
+ template <int dim>
+ std::vector<unsigned int>
+ hermite_lexicographic_to_hierarchic_numbering(const unsigned int regularity)
+ {
+ return Utilities::invert_permutation(
+ hermite_hierarchic_to_lexicographic_numbering<dim>(regularity));
+ }
+
+
+
+ template <int dim>
+ inline std::vector<unsigned int>
+ hermite_face_lexicographic_to_hierarchic_numbering(
+ const unsigned int regularity)
+ {
+ return (dim > 1) ? hermite_lexicographic_to_hierarchic_numbering<dim - 1>(
+ regularity) :
+ std::vector<unsigned int>();
+ }
+
+
+
+ template <int dim>
+ TensorProductPolynomials<dim>
+ get_hermite_polynomials(const unsigned int regularity)
+ {
+ TensorProductPolynomials<dim> poly_space(
+ Polynomials::HermitePoly::generate_complete_basis(regularity));
+
+ std::vector<unsigned int> renumber =
+ internal::hermite_hierarchic_to_lexicographic_numbering<dim>(regularity);
+ poly_space.set_numbering(renumber);
+
+ return poly_space;
+ }
+
+
+
+ static inline unsigned int
+ binomial(const unsigned int n, const unsigned int i)
+ {
+ unsigned int C = 1, k = 1;
+
+ for (unsigned int j = n; j > i; --j)
+ {
+ C *= j;
+ C /= k++;
+ }
+
+ return C;
+ }
+
+
+
+ /**
+ * The @p Rescaler class implements the re-scaling of individual shape
+ * functions required by Hermite bases on non-uniform meshes. The three
+ * cases for different element dimensions are all defined separately
+ * due to the requirement for different levels of nesting of for loops.
+ */
+ template <int xdim, int xspacedim = xdim, typename xNumber = double>
+ class Rescaler
+ {
+ public:
+ void
+ rescale_fe_hermite_values(
+ const FE_Hermite<xdim, xspacedim> & fe_herm,
+ const typename Mapping<xdim, xspacedim>::InternalDataBase &mapping_data,
+ Table<2, xNumber> & value_list);
+
+
+
+ template <int spacedim, typename Number>
+ void
+ rescale_fe_hermite_values(
+ Rescaler<1, spacedim, Number> & /*rescaler*/,
+ const FE_Hermite<1, spacedim> & fe_herm,
+ const typename Mapping<1, spacedim>::InternalDataBase &mapping_data,
+ Table<2, Number> & value_list)
+ {
+ unsigned int n_q_points;
+ double cell_extent = 1.0;
+
+ // Check mapping_data is associated with a compatible mapping class
+ if (dynamic_cast<const typename MappingHermite<1, spacedim>::InternalData
+ *>(&mapping_data) != nullptr)
+ {
+ const typename MappingHermite<1, spacedim>::InternalData *data =
+ dynamic_cast<
+ const typename MappingHermite<1, spacedim>::InternalData *>(
+ &mapping_data);
+ n_q_points = data->quadrature_points.size();
+ cell_extent = data->cell_extents[0];
+ }
+ else if (dynamic_cast<const typename MappingCartesian<1>::InternalData *>(
+ &mapping_data) != nullptr)
+ {
+ const typename MappingCartesian<1>::InternalData *data =
+ dynamic_cast<const typename MappingCartesian<1>::InternalData *>(
+ &mapping_data);
+ n_q_points = data->quadrature_points.size();
+ cell_extent = data->cell_extents[0];
+ }
+ else
+ Assert(false, ExcInternalError());
+
+ const unsigned int regularity = fe_herm.get_regularity();
+ const unsigned int n_dofs_per_cell = fe_herm.n_dofs_per_cell();
+ const unsigned int n_q_points_out = value_list.size(1);
+ (void)n_dofs_per_cell;
+
+ AssertDimension(value_list.size(0), n_dofs_per_cell);
+ AssertDimension(n_dofs_per_cell, 2 * regularity + 2);
+ AssertIndexRange(n_q_points_out, n_q_points + 1);
+ (void)n_q_points;
+
+ std::vector<unsigned int> l2h =
+ hermite_lexicographic_to_hierarchic_numbering<1>(regularity);
+
+ for (unsigned int q = 0; q < n_q_points_out; ++q)
+ {
+ double factor_1 = 1.0;
+
+ for (unsigned int d1 = 0, d2 = regularity + 1; d2 < n_dofs_per_cell;
+ ++d1, ++d2)
+ {
+ /*
+ * d1 is used to count over indices on the left and d2 counts
+ * over indices on the right. These variables are used
+ * to avoid the need to loop over vertices.
+ */
+ value_list(l2h[d1], q) *= factor_1;
+ value_list(l2h[d2], q) *= factor_1;
+
+ factor_1 *= cell_extent;
+ }
+ }
+ }
+
+
+
+ template <int spacedim, typename Number>
+ void
+ rescale_fe_hermite_values(
+ Rescaler<2, spacedim, Number> & /*rescaler*/,
+ const FE_Hermite<2, spacedim> & fe_herm,
+ const typename Mapping<2, spacedim>::InternalDataBase &mapping_data,
+ Table<2, Number> & value_list)
+ {
+ unsigned int n_q_points;
+ Tensor<1, 2> cell_extents;
+
+ // Check mapping_data is associated with a compatible mapping class
+ if (dynamic_cast<const typename MappingHermite<2, spacedim>::InternalData
+ *>(&mapping_data) != nullptr)
+ {
+ const typename MappingHermite<2, spacedim>::InternalData *data =
+ dynamic_cast<
+ const typename MappingHermite<2, spacedim>::InternalData *>(
+ &mapping_data);
+ n_q_points = data->quadrature_points.size();
+ cell_extents = data->cell_extents;
+ }
+ else if (dynamic_cast<const typename MappingCartesian<2>::InternalData *>(
+ &mapping_data) != nullptr)
+ {
+ const typename MappingCartesian<2>::InternalData *data =
+ dynamic_cast<const typename MappingCartesian<2>::InternalData *>(
+ &mapping_data);
+ n_q_points = data->quadrature_points.size();
+ cell_extents = data->cell_extents;
+ }
+ else
+ Assert(false, ExcInternalError());
+
+ const unsigned int regularity = fe_herm.get_regularity();
+ const unsigned int n_dofs_per_cell = fe_herm.n_dofs_per_cell();
+ const unsigned int n_dofs_per_dim = 2 * regularity + 2;
+ const unsigned int n_q_points_out = value_list.size(1);
+ (void)n_dofs_per_cell;
+
+ AssertDimension(value_list.size(0), n_dofs_per_cell);
+ AssertDimension(n_dofs_per_dim * n_dofs_per_dim, n_dofs_per_cell);
+ AssertIndexRange(n_q_points_out, n_q_points + 1);
+ (void)n_q_points;
+
+ std::vector<unsigned int> l2h =
+ hermite_lexicographic_to_hierarchic_numbering<2>(regularity);
+
+ AssertDimension(l2h.size(), n_dofs_per_cell);
+
+ for (unsigned int q = 0; q < n_q_points_out; ++q)
+ {
+ double factor_2 = 1.0;
+
+ for (unsigned int d3 = 0, d4 = regularity + 1; d4 < n_dofs_per_dim;
+ ++d3, ++d4)
+ {
+ double factor_1 = factor_2;
+
+ for (unsigned int d1 = 0, d2 = regularity + 1;
+ d2 < n_dofs_per_dim;
+ ++d1, ++d2)
+ {
+ /*
+ * d1 and d2 represent "left" and "right" in the
+ * x-direction, d3 and d4 represent "bottom" and "top"
+ * in the y-direction. As before, this is to avoid looping
+ * over vertices.
+ */
+ value_list(l2h[d1 + d3 * n_dofs_per_dim], q) *= factor_1;
+ value_list(l2h[d2 + d3 * n_dofs_per_dim], q) *= factor_1;
+ value_list(l2h[d1 + d4 * n_dofs_per_dim], q) *= factor_1;
+ value_list(l2h[d2 + d4 * n_dofs_per_dim], q) *= factor_1;
+
+ factor_1 *= cell_extents[0];
+ }
+
+ factor_2 *= cell_extents[1];
+ }
+ }
+ }
+
+
+
+ template <int spacedim, typename Number>
+ void
+ rescale_fe_hermite_values(
+ Rescaler<3, spacedim, Number> & /*rescaler*/,
+ const FE_Hermite<3, spacedim> & fe_herm,
+ const typename Mapping<3, spacedim>::InternalDataBase &mapping_data,
+ Table<2, Number> & value_list)
+ {
+ unsigned int n_q_points;
+ Tensor<1, 3> cell_extents;
+
+ // Check mapping_data is associated with a compatible mapping class
+ if (dynamic_cast<const typename MappingHermite<3, spacedim>::InternalData
+ *>(&mapping_data) != nullptr)
+ {
+ const typename MappingHermite<3, spacedim>::InternalData *data =
+ dynamic_cast<
+ const typename MappingHermite<3, spacedim>::InternalData *>(
+ &mapping_data);
+ n_q_points = data->quadrature_points.size();
+ cell_extents = data->cell_extents;
+ }
+ else if (dynamic_cast<const typename MappingCartesian<3>::InternalData *>(
+ &mapping_data) != nullptr)
+ {
+ const typename MappingCartesian<3>::InternalData *data =
+ dynamic_cast<const typename MappingCartesian<3>::InternalData *>(
+ &mapping_data);
+ n_q_points = data->quadrature_points.size();
+ cell_extents = data->cell_extents;
+ }
+ else
+ Assert(false, ExcInternalError());
+
+ const unsigned int regularity = fe_herm.get_regularity();
+ const unsigned int n_dofs_per_cell = fe_herm.n_dofs_per_cell();
+ const unsigned int n_dofs_per_dim = 2 * regularity + 2;
+ const unsigned int n_dofs_per_quad = n_dofs_per_dim * n_dofs_per_dim;
+ const unsigned int n_q_points_out = value_list.size(1);
+ (void)n_dofs_per_cell;
+
+ AssertDimension(value_list.size(0), n_dofs_per_cell);
+ AssertDimension(Utilities::pow(n_dofs_per_dim, 3), n_dofs_per_cell);
+ AssertIndexRange(n_q_points_out, n_q_points + 1);
+ (void)n_q_points;
+
+ std::vector<unsigned int> l2h =
+ hermite_lexicographic_to_hierarchic_numbering<3>(regularity);
+
+ for (unsigned int q = 0; q < n_q_points_out; ++q)
+ {
+ double factor_3 = 1.0;
+
+ for (unsigned int d5 = 0, d6 = regularity + 1; d6 < n_dofs_per_dim;
+ ++d5, ++d6)
+ {
+ double factor_2 = factor_3;
+
+ for (unsigned int d3 = 0, d4 = regularity + 1;
+ d4 < n_dofs_per_dim;
+ ++d3, ++d4)
+ {
+ double factor_1 = factor_2;
+
+ for (unsigned int d1 = 0, d2 = regularity + 1;
+ d2 < n_dofs_per_dim;
+ ++d1, ++d2)
+ {
+ /*
+ * d1, d2: "left" and "right" (x-direction)
+ * d3, d4: "bottom" and "top" (y-direction)
+ * d5, d6: "down" and "up" (z-direction)
+ * This avoids looping over vertices
+ */
+ value_list(
+ l2h[d1 + d3 * n_dofs_per_dim + d5 * n_dofs_per_quad],
+ q) *= factor_1;
+ value_list(
+ l2h[d2 + d3 * n_dofs_per_dim + d5 * n_dofs_per_quad],
+ q) *= factor_1;
+ value_list(
+ l2h[d1 + d4 * n_dofs_per_dim + d5 * n_dofs_per_quad],
+ q) *= factor_1;
+ value_list(
+ l2h[d2 + d4 * n_dofs_per_dim + d5 * n_dofs_per_quad],
+ q) *= factor_1;
+ value_list(
+ l2h[d1 + d3 * n_dofs_per_dim + d6 * n_dofs_per_quad],
+ q) *= factor_1;
+ value_list(
+ l2h[d2 + d3 * n_dofs_per_dim + d6 * n_dofs_per_quad],
+ q) *= factor_1;
+ value_list(
+ l2h[d1 + d4 * n_dofs_per_dim + d6 * n_dofs_per_quad],
+ q) *= factor_1;
+ value_list(
+ l2h[d2 + d4 * n_dofs_per_dim + d6 * n_dofs_per_quad],
+ q) *= factor_1;
+
+ factor_1 *= cell_extents[0];
+ }
+
+ factor_2 *= cell_extents[1];
+ }
+
+ factor_3 *= cell_extents[2];
+ }
+ }
+ }
+ }; // class Rescaler
+} // namespace internal
+
+
+
+// Constructors
+template <int dim, int spacedim>
+FE_Hermite<dim, spacedim>::FE_Hermite(const unsigned int regularity)
+ : FE_Poly<dim, spacedim>(
+ internal::get_hermite_polynomials<dim>(regularity),
+ FiniteElementData<dim>(internal::get_hermite_dpo_vector(dim, regularity),
+ 1,
+ 2 * regularity + 1,
+ (regularity > 0 ? FiniteElementData<dim>::H2 :
+
+ FiniteElementData<dim>::H1)),
+ std::vector<bool>(Utilities::pow(2 * (regularity + 1), dim), false),
+ std::vector<ComponentMask>(Utilities::pow(2 * (regularity + 1), dim),
+ std::vector<bool>(1, true)))
+ , regularity(regularity)
+{}
+
+
+
+template <int dim, int spacedim>
+std::string
+FE_Hermite<dim, spacedim>::get_name() const
+{
+ std::ostringstream name_buffer;
+ name_buffer << "FE_Hermite<" << dim << "," << spacedim << ">("
+ << this->regularity << ")";
+ return name_buffer.str();
+}
+
+
+
+template <int dim, int spacedim>
+std::unique_ptr<FiniteElement<dim, spacedim>>
+FE_Hermite<dim, spacedim>::clone() const
+{
+ return std::make_unique<FE_Hermite<dim, spacedim>>(*this);
+}
+
+
+
+template <int dim, int spacedim>
+std::vector<unsigned int>
+FE_Hermite<dim, spacedim>::get_lexicographic_to_hierarchic_numbering() const
+{
+ return internal::hermite_lexicographic_to_hierarchic_numbering<dim>(
+ this->regularity);
+}
+
+
+
+template <int dim, int spacedim>
+void
+FE_Hermite<dim, spacedim>::fill_fe_values(
+ const typename Triangulation<dim, spacedim>::cell_iterator &,
+ const CellSimilarity::Similarity cell_similarity,
+ const Quadrature<dim> & /*quadrature*/,
+ const Mapping<dim, spacedim> & mapping,
+ const typename Mapping<dim, spacedim>::InternalDataBase &mapping_internal,
+ const dealii::internal::FEValuesImplementation::MappingRelatedData<dim,
+ spacedim>
+ & /*mapping_data*/,
+ const typename FiniteElement<dim, spacedim>::InternalDataBase &fe_internal,
+ dealii::internal::FEValuesImplementation::FiniteElementRelatedData<dim,
+ spacedim>
+ &output_data) const
+{
+ // Convert data object to internal data for this class.
+ // Fails with an exception if that is not possible.
+ Assert(
+ (dynamic_cast<const typename FE_Hermite<dim, spacedim>::InternalData *>(
+ &fe_internal) != nullptr),
+ ExcInternalError());
+ const typename FE_Hermite<dim, spacedim>::InternalData &fe_data =
+ static_cast<const typename FE_Hermite<dim, spacedim>::InternalData &>(
+ fe_internal);
+
+ Assert(
+ (dynamic_cast<const typename MappingHermite<dim, spacedim>::InternalData *>(
+ &mapping_internal) != nullptr) ||
+ (dynamic_cast<const typename MappingCartesian<dim>::InternalData *>(
+ &mapping_internal) != nullptr),
+ ExcInternalError());
+
+ const UpdateFlags flags(fe_data.update_each);
+
+ // Transform values gradients and higher derivatives. Values also need to
+ // be rescaled according the the nodal derivative they correspond to.
+ if ((flags & update_values) &&
+ (cell_similarity != CellSimilarity::translation))
+ {
+ internal::Rescaler<dim, spacedim, double> shape_fix;
+ for (unsigned int i = 0; i < output_data.shape_values.size(0); ++i)
+ for (unsigned int q = 0; q < output_data.shape_values.size(1); ++q)
+ output_data.shape_values(i, q) = fe_data.shape_values(i, q);
+ shape_fix.rescale_fe_hermite_values(shape_fix,
+ *this,
+ mapping_internal,
+ output_data.shape_values);
+ }
+
+ if ((flags & update_gradients) &&
+ (cell_similarity != CellSimilarity::translation))
+ {
+ for (unsigned int k = 0; k < this->n_dofs_per_cell(); ++k)
+ mapping.transform(make_array_view(fe_data.shape_gradients, k),
+ mapping_covariant,
+ mapping_internal,
+ make_array_view(output_data.shape_gradients, k));
+
+ internal::Rescaler<dim, spacedim, Tensor<1, spacedim>> grad_fix;
+ grad_fix.rescale_fe_hermite_values(grad_fix,
+ *this,
+ mapping_internal,
+ output_data.shape_gradients);
+ }
+
+ if ((flags & update_hessians) &&
+ (cell_similarity != CellSimilarity::translation))
+ {
+ for (unsigned int k = 0; k < this->n_dofs_per_cell(); ++k)
+ mapping.transform(make_array_view(fe_data.shape_hessians, k),
+ mapping_covariant_gradient,
+ mapping_internal,
+ make_array_view(output_data.shape_hessians, k));
+
+ internal::Rescaler<dim, spacedim, Tensor<2, spacedim>> hessian_fix;
+ hessian_fix.rescale_fe_hermite_values(hessian_fix,
+ *this,
+ mapping_internal,
+ output_data.shape_hessians);
+ }
+
+ if ((flags & update_3rd_derivatives) &&
+ (cell_similarity != CellSimilarity::translation))
+ {
+ for (unsigned int k = 0; k < this->n_dofs_per_cell(); ++k)
+ mapping.transform(make_array_view(fe_data.shape_3rd_derivatives, k),
+ mapping_covariant_hessian,
+ mapping_internal,
+ make_array_view(output_data.shape_3rd_derivatives,
+ k));
+
+ internal::Rescaler<dim, spacedim, Tensor<3, spacedim>> third_dev_fix;
+ third_dev_fix.rescale_fe_hermite_values(
+ third_dev_fix,
+ *this,
+ mapping_internal,
+ output_data.shape_3rd_derivatives);
+ }
+}
+
+
+
+template <int dim, int spacedim>
+void
+FE_Hermite<dim, spacedim>::fill_fe_face_values(
+ const typename Triangulation<dim, spacedim>::cell_iterator &cell,
+ const unsigned int face_no,
+ const hp::QCollection<dim - 1> & quadrature,
+ const Mapping<dim, spacedim> & mapping,
+ const typename Mapping<dim, spacedim>::InternalDataBase & mapping_internal,
+ const dealii::internal::FEValuesImplementation::MappingRelatedData<dim,
+ spacedim>
+ &,
+ const typename FiniteElement<dim, spacedim>::InternalDataBase &fe_internal,
+ dealii::internal::FEValuesImplementation::FiniteElementRelatedData<dim,
+ spacedim>
+ &output_data) const
+{
+ /*
+ * Convert data object to internal data for this class. Fails with
+ * an exception if that is not possible.
+ */
+ Assert(
+ (dynamic_cast<const typename FE_Hermite<dim, spacedim>::InternalData *>(
+ &fe_internal) != nullptr),
+ ExcInternalError());
+ const typename FE_Hermite<dim, spacedim>::InternalData &fe_data =
+ static_cast<const typename FE_Hermite<dim, spacedim>::InternalData &>(
+ fe_internal);
+
+ Assert(
+ (dynamic_cast<const typename MappingHermite<dim, spacedim>::InternalData *>(
+ &mapping_internal) != nullptr) ||
+ (dynamic_cast<const typename MappingCartesian<dim, spacedim>::InternalData
+ *>(&mapping_internal) != nullptr),
+ ExcInternalError());
+
+ AssertDimension(quadrature.size(), 1U);
+
+ /*
+ * offset determines which data set to take (all data sets for all
+ * faces are stored contiguously)
+ */
+ const typename QProjector<dim>::DataSetDescriptor offset =
+ QProjector<dim>::DataSetDescriptor::face(face_no,
+ cell->face_orientation(face_no),
+ cell->face_flip(face_no),
+ cell->face_rotation(face_no),
+ quadrature[0].size());
+
+ const UpdateFlags flags(fe_data.update_each);
+
+ // Transform values, gradients and higher derivatives.
+ if (flags & update_values)
+ {
+ for (unsigned int k = 0; k < this->dofs_per_cell; ++k)
+ for (unsigned int i = 0; i < quadrature[0].size(); ++i)
+ output_data.shape_values(k, i) = fe_data.shape_values[k][i + offset];
+
+ internal::Rescaler<dim, spacedim, double> shape_face_fix;
+ shape_face_fix.rescale_fe_hermite_values(shape_face_fix,
+ *this,
+ mapping_internal,
+ output_data.shape_values);
+ }
+
+ if (flags & update_gradients)
+ {
+ for (unsigned int k = 0; k < this->dofs_per_cell; ++k)
+ mapping.transform(make_array_view(fe_data.shape_gradients,
+ k,
+ offset,
+ quadrature[0].size()),
+ mapping_covariant,
+ mapping_internal,
+ make_array_view(output_data.shape_gradients, k));
+
+ internal::Rescaler<dim, spacedim, Tensor<1, spacedim>> grad_face_fix;
+ grad_face_fix.rescale_fe_hermite_values(grad_face_fix,
+ *this,
+ mapping_internal,
+ output_data.shape_gradients);
+ }
+
+ if (flags & update_hessians)
+ {
+ for (unsigned int k = 0; k < this->dofs_per_cell; ++k)
+ mapping.transform(make_array_view(fe_data.shape_hessians,
+ k,
+ offset,
+ quadrature[0].size()),
+ mapping_covariant_gradient,
+ mapping_internal,
+ make_array_view(output_data.shape_hessians, k));
+
+ internal::Rescaler<dim, spacedim, Tensor<2, spacedim>> hessian_face_fix;
+ hessian_face_fix.rescale_fe_hermite_values(hessian_face_fix,
+ *this,
+ mapping_internal,
+ output_data.shape_hessians);
+ }
+
+ if (flags & update_3rd_derivatives)
+ {
+ for (unsigned int k = 0; k < this->dofs_per_cell; ++k)
+ mapping.transform(make_array_view(fe_data.shape_3rd_derivatives,
+ k,
+ offset,
+ quadrature[0].size()),
+ mapping_covariant_hessian,
+ mapping_internal,
+ make_array_view(output_data.shape_3rd_derivatives,
+ k));
+
+ internal::Rescaler<dim, spacedim, Tensor<3, spacedim>> shape_3rd_face_fix;
+ shape_3rd_face_fix.rescale_fe_hermite_values(
+ shape_3rd_face_fix,
+ *this,
+ mapping_internal,
+ output_data.shape_3rd_derivatives);
+ }
+}
+
+
+
+// Explicit instantiations
+#include "fe_hermite.inst"
+
+DEAL_II_NAMESPACE_CLOSE
--- /dev/null
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2021 - 2022 by the deal.II authors
+//
+// This file is part of the deal.II library.
+//
+// The deal.II library is free software; you can use it, redistribute
+// it, and/or modify it under the terms of the GNU Lesser General
+// Public License as published by the Free Software Foundation; either
+// version 2.1 of the License, or (at your option) any later version.
+// The full text of the license can be found in the file LICENSE.md at
+// the top level directory of deal.II.
+//
+// ---------------------------------------------------------------------
+
+
+
+for (deal_II_dimension : DIMENSIONS)
+ {
+ template void
+ internal::hermite_hierarchic_to_lexicographic_numbering<deal_II_dimension>(
+ const unsigned int regularity, std::vector<unsigned int> &h2l);
+
+ template std::vector<unsigned int>
+ internal::hermite_lexicographic_to_hierarchic_numbering<deal_II_dimension>(
+ const unsigned int regularity);
+
+ template std::vector<unsigned int> internal::
+ hermite_face_lexicographic_to_hierarchic_numbering<deal_II_dimension>(
+ const unsigned int regularity);
+ }
+
+
+
+for (deal_II_dimension : DIMENSIONS; deal_II_space_dimension : SPACE_DIMENSIONS;
+ tensor_rank : RANKS)
+ {
+#if deal_II_dimension <= deal_II_space_dimension
+ template class internal::Rescaler<
+ deal_II_dimension,
+ deal_II_space_dimension,
+ Tensor<tensor_rank, deal_II_space_dimension>>;
+#endif
+ }
+
+
+
+for (deal_II_dimension : DIMENSIONS; deal_II_space_dimension : SPACE_DIMENSIONS)
+ {
+#if deal_II_dimension <= deal_II_space_dimension
+ template class internal::Rescaler<deal_II_dimension,
+ deal_II_space_dimension>;
+ template class FE_Hermite<deal_II_dimension, deal_II_space_dimension>;
+#endif
+ }
solution_transfer_inst2.cc
solution_transfer_inst3.cc
solution_transfer_inst4.cc
+ vector_tools_hermite.cc
vector_tools_integrate_difference.cc
vector_tools_interpolate.cc
vector_tools_point_value.cc
time_dependent.inst.in
vector_tools_boundary.inst.in
vector_tools_constraints.inst.in
+ vector_tools_hermite.inst.in
vector_tools_integrate_difference.inst.in
vector_tools_interpolate.inst.in
vector_tools_mean_value.inst.in
--- /dev/null
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2021 - 2022 by the deal.II authors
+//
+// This file is part of the deal.II library.
+//
+// The deal.II library is free software; you can use it, redistribute
+// it, and/or modify it under the terms of the GNU Lesser General
+// Public License as published by the Free Software Foundation; either
+// version 2.1 of the License, or (at your option) any later version.
+// The full text of the license can be found in the file LICENSE.md at
+// the top level directory of deal.II.
+//
+// ---------------------------------------------------------------------
+
+#include <deal.II/numerics/vector_tools_hermite.templates.h>
+
+DEAL_II_NAMESPACE_OPEN
+
+// explicit instantiations
+#include "vector_tools_hermite.inst"
+
+DEAL_II_NAMESPACE_CLOSE
--- /dev/null
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2021 - 2022 by the deal.II authors
+//
+// This file is part of the deal.II library.
+//
+// The deal.II library is free software; you can use it, redistribute
+// it, and/or modify it under the terms of the GNU Lesser General
+// Public License as published by the Free Software Foundation; either
+// version 2.1 of the License, or (at your option) any later version.
+// The full text of the license can be found in the file LICENSE.md at
+// the top level directory of deal.II.
+//
+// ---------------------------------------------------------------------
+
+
+for (deal_II_dimension : DIMENSIONS)
+ {
+ template void VectorTools::internal::do_hermite_direct_projection<
+ deal_II_dimension,
+ deal_II_dimension,
+ double>(const Mapping<deal_II_dimension> & mapping_h,
+ const DoFHandler<deal_II_dimension> &dof_handler,
+ const std::map<types::boundary_id,
+ const Function<deal_II_dimension, double> *>
+ & boundary_functions,
+ const Quadrature<deal_II_dimension - 1> & quadrature,
+ const unsigned int position,
+ std::map<types::global_dof_index, double> &boundary_values,
+ std::vector<unsigned int> component_mapping);
+
+ template void VectorTools::project_hermite_boundary_values<
+ deal_II_dimension,
+ deal_II_dimension,
+ double>(const Mapping<deal_II_dimension> & mapping_h,
+ const DoFHandler<deal_II_dimension> &dof_handler,
+ const std::map<types::boundary_id,
+ const Function<deal_II_dimension, double> *>
+ & boundary_functions,
+ const Quadrature<deal_II_dimension - 1> & quadrature,
+ const HermiteBoundaryType projection_mode,
+ std::map<types::global_dof_index, double> &boundary_values,
+ std::vector<unsigned int> component_mapping);
+
+ template void VectorTools::project_hermite_boundary_values<
+ deal_II_dimension,
+ deal_II_dimension,
+ double>(const Mapping<deal_II_dimension> & mapping_h,
+ const DoFHandler<deal_II_dimension> &dof_handler,
+ const std::map<types::boundary_id,
+ const Function<deal_II_dimension, double> *>
+ & boundary_functions,
+ const Quadrature<deal_II_dimension - 1> & quadrature,
+ std::map<types::global_dof_index, double> &boundary_values,
+ std::vector<unsigned int> component_mapping);
+
+ template void VectorTools::
+ project_hermite<deal_II_dimension, Vector<double>, deal_II_dimension>(
+ const Mapping<deal_II_dimension> & mapping,
+ const DoFHandler<deal_II_dimension> & dof,
+ const AffineConstraints<Vector<double>::value_type> &constraints,
+ const Quadrature<deal_II_dimension> & quadrature,
+ const Function<deal_II_dimension, Vector<double>::value_type> &function,
+ Vector<double> & vec,
+ const bool enforce_zero_boundary,
+ const Quadrature<deal_II_dimension - 1> &q_boundary,
+ const bool project_to_boundary_first);
+ }
--- /dev/null
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2021 - 2022 by the deal.II authors
+//
+// This file is part of the deal.II library.
+//
+// The deal.II library is free software; you can use it, redistribute
+// it, and/or modify it under the terms of the GNU Lesser General
+// Public License as published by the Free Software Foundation; either
+// version 2.1 of the License, or (at your option) any later version.
+// The full text of the license can be found in the file LICENSE.md at
+// the top level directory of deal.II.
+//
+// ---------------------------------------------------------------------
+
+#include <deal.II/base/config.h>
+
+#include <deal.II/base/function.h>
+#include <deal.II/base/quadrature_lib.h>
+#include <deal.II/base/utilities.h>
+
+#include <deal.II/dofs/dof_accessor.h>
+#include <deal.II/dofs/dof_handler.h>
+#include <deal.II/dofs/dof_tools.h>
+
+#include <deal.II/fe/fe_hermite.h>
+#include <deal.II/fe/fe_interface_values.h>
+#include <deal.II/fe/fe_values.h>
+#include <deal.II/fe/mapping_cartesian.h>
+
+#include <deal.II/grid/grid_generator.h>
+#include <deal.II/grid/tria.h>
+#include <deal.II/grid/tria_accessor.h>
+#include <deal.II/grid/tria_iterator.h>
+
+#include <deal.II/lac/affine_constraints.h>
+#include <deal.II/lac/dynamic_sparsity_pattern.h>
+#include <deal.II/lac/full_matrix.h>
+#include <deal.II/lac/precondition.h>
+#include <deal.II/lac/solver_cg.h>
+#include <deal.II/lac/sparse_matrix.h>
+#include <deal.II/lac/vector.h>
+
+#include <deal.II/numerics/data_out.h>
+#include <deal.II/numerics/matrix_tools.h>
+#include <deal.II/numerics/vector_tools.h>
+
+#include <cmath>
+#include <cstdlib>
+#include <fstream>
+#include <iterator>
+#include <sstream>
+#include <vector>
+
+
+
+/*
+ * Test case for Hermite on an irregular 1D grid.
+ * <code>FE_Hermite<dim>(reg)<\code> should be able to perfectly represent any
+ * polynomial function up to degree $2 \times reg+1$. If all basis functions
+ * are correctly scaled according to element size, then projecting a
+ * polynomial of this form onto the FE space will produce negligible pointwise
+ * errors.
+ */
+
+using namespace dealii;
+
+// Define a function to project onto the domain [-1,1]^dim
+template <int dim>
+class Solution : public Function<dim>
+{
+public:
+ virtual double
+ value(const Point<dim> &p, unsigned int c = 0) const override
+ {
+ (void)c;
+ switch (dim)
+ {
+ case 2:
+ return p(0) * p(1);
+ case 1:
+ case 3:
+ default:
+ Assert(false, ExcNotImplemented());
+ return 0;
+ }
+ }
+
+
+
+ std::string
+ get_function_string()
+ {
+ switch (dim)
+ {
+ case 2:
+ return "XY";
+ case 1:
+ case 3:
+ default:
+ Assert(false, ExcNotImplemented());
+ return "";
+ }
+ }
+};
+
+
+
+template <int dim>
+void
+test_fe_on_domain(const unsigned int regularity)
+{
+ deallog << std::endl;
+ char fname[50];
+ sprintf(fname, "Cell-%dd-Hermite-%d", dim, regularity);
+ deallog.push(fname);
+ deallog.depth_file(2);
+
+ Triangulation<dim> tr;
+ DoFHandler<dim> dof(tr);
+
+ double left = -1.0, right = 1.0;
+ Point<dim> left_point, right_point;
+ for (unsigned int i = 0; i < dim; ++i)
+ left_point(i) = left, right_point(i) = right;
+ GridGenerator::subdivided_hyper_cube(tr, 4, left, right);
+
+ FE_Hermite<dim> herm(regularity);
+ dof.distribute_dofs(herm);
+
+ MappingCartesian<dim> mapping;
+
+ QGauss<dim> quadr(2 * regularity + 2);
+
+ Vector<double> sol(dof.n_dofs());
+ Vector<double> rhs(dof.n_dofs());
+
+ Solution<dim> sol_object;
+
+ AffineConstraints<double> constraints;
+ constraints.close();
+
+ DynamicSparsityPattern dsp(dof.n_dofs());
+ DoFTools::make_sparsity_pattern(dof, dsp);
+ SparsityPattern sp;
+ sp.copy_from(dsp);
+
+ SparseMatrix<double> mass_matrix;
+ mass_matrix.reinit(sp);
+ MatrixCreator::create_mass_matrix(mapping, dof, quadr, mass_matrix);
+
+ FEValues<dim> fe_herm(mapping,
+ herm,
+ quadr,
+ update_values | update_quadrature_points |
+ update_JxW_values);
+
+ std::vector<types::global_dof_index> local_to_global(herm.n_dofs_per_cell());
+
+ for (const auto &cell : dof.active_cell_iterators())
+ {
+ fe_herm.reinit(cell);
+ cell->get_dof_indices(local_to_global);
+ for (const unsigned int i : fe_herm.dof_indices())
+ {
+ double rhs_temp = 0;
+ for (const unsigned int q : fe_herm.quadrature_point_indices())
+ rhs_temp += fe_herm.shape_value(i, q) *
+ sol_object.value(fe_herm.quadrature_point(q)) *
+ fe_herm.JxW(q);
+ rhs(local_to_global[i]) += rhs_temp;
+ }
+ }
+
+ IterationNumberControl solver_control_values(8000, 1e-11);
+ SolverCG<> solver(solver_control_values);
+
+ solver.solve(mass_matrix, sol, rhs, PreconditionIdentity());
+
+ double err_sq = 0;
+
+ for (auto &cell : dof.active_cell_iterators())
+ {
+ fe_herm.reinit(cell);
+ cell->get_dof_indices(local_to_global);
+ for (const unsigned int q : fe_herm.quadrature_point_indices())
+ {
+ double sol_at_point = 0;
+ for (const unsigned int i : fe_herm.dof_indices())
+ sol_at_point += fe_herm.shape_value(i, q) * sol(local_to_global[i]);
+
+ sol_at_point -= sol_object.value(fe_herm.quadrature_point(q));
+ err_sq += sol_at_point * sol_at_point * fe_herm.JxW(q);
+ }
+ }
+
+ err_sq = std::sqrt(err_sq);
+
+ deallog << "Test polynomial:" << std::endl;
+ deallog << sol_object.get_function_string() << std::endl;
+ deallog << std::endl;
+
+ deallog << "Interpolation error:" << std::endl;
+ deallog << err_sq << "\n\n" << std::endl;
+ deallog.pop();
+}
+
+
+
+int
+main()
+{
+ std::ofstream logfile("output");
+ deallog << std::setprecision(8) << std::fixed;
+ deallog.attach(logfile);
+ MultithreadInfo::set_thread_limit(1);
+
+ test_fe_on_domain<2>(0);
+ test_fe_on_domain<2>(1);
+ test_fe_on_domain<2>(2);
+
+ return 0;
+}
--- /dev/null
+
+DEAL::
+DEAL:Cell-2d-Hermite-0::Test polynomial:
+DEAL:Cell-2d-Hermite-0::XY
+DEAL:Cell-2d-Hermite-0::
+DEAL:Cell-2d-Hermite-0::Interpolation error:
+DEAL:Cell-2d-Hermite-0::0.00000000
+
+
+DEAL::
+DEAL:Cell-2d-Hermite-1::Test polynomial:
+DEAL:Cell-2d-Hermite-1::XY
+DEAL:Cell-2d-Hermite-1::
+DEAL:Cell-2d-Hermite-1::Interpolation error:
+DEAL:Cell-2d-Hermite-1::0.00000000
+
+
+DEAL::
+DEAL:Cell-2d-Hermite-2::Test polynomial:
+DEAL:Cell-2d-Hermite-2::XY
+DEAL:Cell-2d-Hermite-2::
+DEAL:Cell-2d-Hermite-2::Interpolation error:
+DEAL:Cell-2d-Hermite-2::0.00000000
+
+
--- /dev/null
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2021 - 2022 by the deal.II authors
+//
+// This file is part of the deal.II library.
+//
+// The deal.II library is free software; you can use it, redistribute
+// it, and/or modify it under the terms of the GNU Lesser General
+// Public License as published by the Free Software Foundation; either
+// version 2.1 of the License, or (at your option) any later version.
+// The full text of the license can be found in the file LICENSE.md at
+// the top level directory of deal.II.
+//
+// ---------------------------------------------------------------------
+
+#ifndef DERIVATIVE_TEST_HEADER
+#define DERIVATIVE_TEST_HEADER
+
+#include <deal.II/base/quadrature_lib.h>
+
+#include <deal.II/dofs/dof_accessor.h>
+
+#include <deal.II/fe/fe_values.h>
+#include <deal.II/fe/mapping_q1.h>
+
+#include <deal.II/grid/grid_generator.h>
+#include <deal.II/grid/tria.h>
+#include <deal.II/grid/tria_iterator.h>
+
+#include <deal.II/lac/vector.h>
+
+#include <cmath>
+#include <fstream>
+#include <string>
+#include <vector>
+
+#include "../tests.h"
+
+char fname[50];
+
+
+
+//////////////////////////////////////////////////////////////////////////////////////////////////////
+// Plot derivatives of the shape functions at the corner nodes of [0,1]^d
+//
+// Output values on each line take the form
+//
+// x (y) (z) xderiv_index (yderiv_index) (zderiv_index) value[0]+1.
+// value[1]+1. ...
+//////////////////////////////////////////////////////////////////////////////////////////////////////
+
+template <int dim>
+inline void
+plot_function_derivatives(Mapping<dim> & mapping,
+ FiniteElement<dim> &finel,
+ const char * name)
+{
+ Triangulation<dim> tr;
+ DoFHandler<dim> dof(tr);
+ GridGenerator::hyper_cube(tr, 0., 1.);
+ dof.distribute_dofs(finel);
+
+ typename DoFHandler<dim>::cell_iterator c = dof.begin();
+
+ const unsigned int sz = finel.n_dofs_per_cell();
+ const unsigned int degree = finel.tensor_degree();
+ const unsigned int max_test_deriv =
+ std::min(((dim == 1) ? 3 : 1), static_cast<int>(degree - 1) / 2);
+ const unsigned int div = 1;
+
+ QTrapez<dim> q;
+ FEValues<dim> fe(mapping,
+ finel,
+ q,
+ UpdateFlags(update_values | update_gradients |
+ update_hessians | update_3rd_derivatives));
+
+ sprintf(fname, "Cell-%dd-%s", dim, name);
+ deallog.push(fname);
+
+ fe.reinit(c);
+
+ unsigned int k = 0;
+
+ // iterate over vertices
+ for (unsigned int mz = 0; mz <= ((dim > 2) ? div : 0); ++mz)
+ {
+ for (unsigned int my = 0; my <= ((dim > 1) ? div : 0); ++my)
+ {
+ for (unsigned int mx = 0; mx <= div; ++mx)
+ {
+ // iterate over derivatives
+ for (unsigned int dz = 0; dz <= ((dim > 2) ? max_test_deriv : 0);
+ ++dz)
+ {
+ for (unsigned int dy = 0;
+ dy <= ((dim > 1) ? max_test_deriv : 0);
+ ++dy)
+ {
+ for (unsigned int dx = 0; dx <= max_test_deriv; ++dx)
+ {
+ deallog << q.point(k) << " " << dx;
+ if (dim > 1)
+ deallog << " " << dy;
+ if (dim > 2)
+ deallog << " " << dz;
+
+ for (unsigned int i = 0; i < sz; ++i)
+ {
+ unsigned int deriv_level = dx + dy + dz;
+ unsigned int entry_1, entry_2, entry_3;
+
+ switch (deriv_level)
+ {
+ case 0:
+ deallog << " " << fe.shape_value(i, k) + 1.;
+ break;
+
+ case 1:
+ entry_1 = dy + 2 * dz;
+ deallog
+ << " "
+ << fe.shape_grad(i, k)[entry_1] + 1.;
+ break;
+
+ case 2:
+ entry_1 = (dx == 0) ? 1 : 0;
+ entry_2 = 2 * dz - dy * dz + dy;
+ deallog
+ << " "
+ << fe.shape_hessian(i,
+ k)[entry_1][entry_2] +
+ 1.;
+ break;
+
+ case 3:
+ if (dz == 1)
+ entry_1 = 0, entry_2 = 1,
+ entry_3 = 2; // only occurs for dx=dy=dz=1
+ else if (dx * dy == 0)
+ entry_1 = entry_2 = entry_3 = dy / 3;
+ else
+ entry_1 = 0, entry_2 = dy - 1,
+ entry_3 = 1;
+ deallog
+ << " "
+ << fe.shape_3rd_derivative(
+ i, k)[entry_1][entry_2][entry_3] +
+ 1.;
+ break;
+
+ default:
+ ExcMessage(
+ "Requested order of derivative is too high for current implementation.");
+ break;
+ }
+ }
+ deallog << std::endl;
+ }
+ }
+ }
+ deallog << std::endl;
+ ++k;
+ }
+ deallog << std::endl;
+ }
+ deallog << std::endl;
+ }
+ deallog.pop();
+}
+
+#endif
--- /dev/null
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2021 - 2022 by the deal.II authors
+//
+// This file is part of the deal.II library.
+//
+// The deal.II library is free software; you can use it, redistribute
+// it, and/or modify it under the terms of the GNU Lesser General
+// Public License as published by the Free Software Foundation; either
+// version 2.1 of the License, or (at your option) any later version.
+// The full text of the license can be found in the file LICENSE.md at
+// the top level directory of deal.II.
+//
+// ---------------------------------------------------------------------
+
+#include <deal.II/fe/fe_hermite.h>
+#include <deal.II/fe/mapping_cartesian.h>
+
+#include <string>
+
+#include "../tests.h"
+
+#include "derivatives.h"
+
+#define PRECISION 8
+
+/*
+ * Tests the values of derivatives of an Hermite basis at the boundaries
+ * of an element in 1D. This is important to ensure that the basis is
+ * behaving as expected, and will not accidentally enforce discontinuities
+ * in derivatives across element boundaries.
+ */
+
+
+template <int dim>
+void
+print_hermite_endpoint_derivatives()
+{
+ MappingCartesian<dim> m;
+
+ FE_Hermite<dim> herm0(0);
+ plot_function_derivatives<dim>(m, herm0, "Hermite-0");
+
+ FE_Hermite<dim> herm1(1);
+ plot_function_derivatives<dim>(m, herm1, "Hermite-1");
+
+ // Skip the following for dim 3 or greater
+ if (dim < 3)
+ {
+ FE_Hermite<dim> herm2(2);
+ plot_function_derivatives<dim>(m, herm2, "Hermite-2");
+ }
+ if (dim == 1)
+ {
+ FE_Hermite<dim> herm3(3);
+ plot_function_derivatives<dim>(m, herm3, "Hermite-3");
+
+ FE_Hermite<dim> herm4(4);
+ plot_function_derivatives<dim>(m, herm4, "Hermite-4");
+ }
+}
+
+
+
+int
+main()
+{
+ std::ofstream logfile("output");
+
+ deallog << std::setprecision(PRECISION) << std::fixed;
+ deallog.attach(logfile);
+
+ print_hermite_endpoint_derivatives<1>();
+ print_hermite_endpoint_derivatives<2>();
+ print_hermite_endpoint_derivatives<3>();
+
+ return 0;
+}
--- /dev/null
+
+DEAL:Cell-1d-Hermite-0::0.00000000 0 2.00000000 1.00000000
+DEAL:Cell-1d-Hermite-0::
+DEAL:Cell-1d-Hermite-0::1.00000000 0 1.00000000 2.00000000
+DEAL:Cell-1d-Hermite-0::
+DEAL:Cell-1d-Hermite-0::
+DEAL:Cell-1d-Hermite-0::
+DEAL:Cell-1d-Hermite-1::0.00000000 0 2.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-1d-Hermite-1::0.00000000 1 1.00000000 5.00000000 1.00000000 1.00000000
+DEAL:Cell-1d-Hermite-1::
+DEAL:Cell-1d-Hermite-1::1.00000000 0 1.00000000 1.00000000 2.00000000 1.00000000
+DEAL:Cell-1d-Hermite-1::1.00000000 1 1.00000000 1.00000000 1.00000000 5.00000000
+DEAL:Cell-1d-Hermite-1::
+DEAL:Cell-1d-Hermite-1::
+DEAL:Cell-1d-Hermite-1::
+DEAL:Cell-1d-Hermite-2::0.00000000 0 2.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-1d-Hermite-2::0.00000000 1 1.00000000 5.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-1d-Hermite-2::0.00000000 2 1.00000000 1.00000000 33.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-1d-Hermite-2::
+DEAL:Cell-1d-Hermite-2::1.00000000 0 1.00000000 1.00000000 1.00000000 2.00000000 1.00000000 1.00000000
+DEAL:Cell-1d-Hermite-2::1.00000000 1 1.00000000 1.00000000 1.00000000 1.00000000 5.00000000 1.00000000
+DEAL:Cell-1d-Hermite-2::1.00000000 2 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 33.00000000
+DEAL:Cell-1d-Hermite-2::
+DEAL:Cell-1d-Hermite-2::
+DEAL:Cell-1d-Hermite-2::
+DEAL:Cell-1d-Hermite-3::0.00000000 0 2.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-1d-Hermite-3::0.00000000 1 1.00000000 5.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-1d-Hermite-3::0.00000000 2 1.00000000 1.00000000 33.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-1d-Hermite-3::0.00000000 3 1.00000000 1.00000000 1.00000000 385.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-1d-Hermite-3::
+DEAL:Cell-1d-Hermite-3::1.00000000 0 1.00000000 1.00000000 1.00000000 1.00000000 2.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-1d-Hermite-3::1.00000000 1 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 5.00000000 1.00000000 1.00000000
+DEAL:Cell-1d-Hermite-3::1.00000000 2 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 33.00000000 1.00000000
+DEAL:Cell-1d-Hermite-3::1.00000000 3 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 385.00000000
+DEAL:Cell-1d-Hermite-3::
+DEAL:Cell-1d-Hermite-3::
+DEAL:Cell-1d-Hermite-3::
+DEAL:Cell-1d-Hermite-4::0.00000000 0 2.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-1d-Hermite-4::0.00000000 1 1.00000000 5.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-1d-Hermite-4::0.00000000 2 1.00000000 1.00000000 33.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-1d-Hermite-4::0.00000000 3 1.00000000 1.00000000 1.00000000 385.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-1d-Hermite-4::
+DEAL:Cell-1d-Hermite-4::1.00000000 0 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 2.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-1d-Hermite-4::1.00000000 1 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 5.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-1d-Hermite-4::1.00000000 2 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 33.00000000 1.00000000 1.00000000
+DEAL:Cell-1d-Hermite-4::1.00000000 3 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 385.00000000 1.00000000
+DEAL:Cell-1d-Hermite-4::
+DEAL:Cell-1d-Hermite-4::
+DEAL:Cell-1d-Hermite-4::
+DEAL:Cell-2d-Hermite-0::0.00000000 0.00000000 0 0 2.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-2d-Hermite-0::
+DEAL:Cell-2d-Hermite-0::1.00000000 0.00000000 0 0 1.00000000 2.00000000 1.00000000 1.00000000
+DEAL:Cell-2d-Hermite-0::
+DEAL:Cell-2d-Hermite-0::
+DEAL:Cell-2d-Hermite-0::0.00000000 1.00000000 0 0 1.00000000 1.00000000 2.00000000 1.00000000
+DEAL:Cell-2d-Hermite-0::
+DEAL:Cell-2d-Hermite-0::1.00000000 1.00000000 0 0 1.00000000 1.00000000 1.00000000 2.00000000
+DEAL:Cell-2d-Hermite-0::
+DEAL:Cell-2d-Hermite-0::
+DEAL:Cell-2d-Hermite-0::
+DEAL:Cell-2d-Hermite-1::0.00000000 0.00000000 0 0 2.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-2d-Hermite-1::0.00000000 0.00000000 1 0 1.00000000 5.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-2d-Hermite-1::0.00000000 0.00000000 0 1 1.00000000 1.00000000 5.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-2d-Hermite-1::0.00000000 0.00000000 1 1 1.00000000 1.00000000 1.00000000 17.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-2d-Hermite-1::
+DEAL:Cell-2d-Hermite-1::1.00000000 0.00000000 0 0 1.00000000 1.00000000 1.00000000 1.00000000 2.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-2d-Hermite-1::1.00000000 0.00000000 1 0 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 5.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-2d-Hermite-1::1.00000000 0.00000000 0 1 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 5.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-2d-Hermite-1::1.00000000 0.00000000 1 1 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 17.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-2d-Hermite-1::
+DEAL:Cell-2d-Hermite-1::
+DEAL:Cell-2d-Hermite-1::0.00000000 1.00000000 0 0 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 2.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-2d-Hermite-1::0.00000000 1.00000000 1 0 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 5.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-2d-Hermite-1::0.00000000 1.00000000 0 1 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 5.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-2d-Hermite-1::0.00000000 1.00000000 1 1 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 17.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-2d-Hermite-1::
+DEAL:Cell-2d-Hermite-1::1.00000000 1.00000000 0 0 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 2.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-2d-Hermite-1::1.00000000 1.00000000 1 0 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 5.00000000 1.00000000 1.00000000
+DEAL:Cell-2d-Hermite-1::1.00000000 1.00000000 0 1 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 5.00000000 1.00000000
+DEAL:Cell-2d-Hermite-1::1.00000000 1.00000000 1 1 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 17.00000000
+DEAL:Cell-2d-Hermite-1::
+DEAL:Cell-2d-Hermite-1::
+DEAL:Cell-2d-Hermite-1::
+DEAL:Cell-2d-Hermite-2::0.00000000 0.00000000 0 0 2.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-2d-Hermite-2::0.00000000 0.00000000 1 0 1.00000000 5.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-2d-Hermite-2::0.00000000 0.00000000 0 1 1.00000000 1.00000000 1.00000000 5.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-2d-Hermite-2::0.00000000 0.00000000 1 1 1.00000000 1.00000000 1.00000000 1.00000000 17.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-2d-Hermite-2::
+DEAL:Cell-2d-Hermite-2::1.00000000 0.00000000 0 0 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 2.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-2d-Hermite-2::1.00000000 0.00000000 1 0 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 5.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-2d-Hermite-2::1.00000000 0.00000000 0 1 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 5.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-2d-Hermite-2::1.00000000 0.00000000 1 1 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 17.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-2d-Hermite-2::
+DEAL:Cell-2d-Hermite-2::
+DEAL:Cell-2d-Hermite-2::0.00000000 1.00000000 0 0 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 2.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-2d-Hermite-2::0.00000000 1.00000000 1 0 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 5.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-2d-Hermite-2::0.00000000 1.00000000 0 1 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 5.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-2d-Hermite-2::0.00000000 1.00000000 1 1 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 17.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-2d-Hermite-2::
+DEAL:Cell-2d-Hermite-2::1.00000000 1.00000000 0 0 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 2.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-2d-Hermite-2::1.00000000 1.00000000 1 0 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 5.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-2d-Hermite-2::1.00000000 1.00000000 0 1 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 5.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-2d-Hermite-2::1.00000000 1.00000000 1 1 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 17.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-2d-Hermite-2::
+DEAL:Cell-2d-Hermite-2::
+DEAL:Cell-2d-Hermite-2::
+DEAL:Cell-3d-Hermite-0::0.00000000 0.00000000 0.00000000 0 0 0 2.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-3d-Hermite-0::
+DEAL:Cell-3d-Hermite-0::1.00000000 0.00000000 0.00000000 0 0 0 1.00000000 2.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-3d-Hermite-0::
+DEAL:Cell-3d-Hermite-0::
+DEAL:Cell-3d-Hermite-0::0.00000000 1.00000000 0.00000000 0 0 0 1.00000000 1.00000000 2.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-3d-Hermite-0::
+DEAL:Cell-3d-Hermite-0::1.00000000 1.00000000 0.00000000 0 0 0 1.00000000 1.00000000 1.00000000 2.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-3d-Hermite-0::
+DEAL:Cell-3d-Hermite-0::
+DEAL:Cell-3d-Hermite-0::
+DEAL:Cell-3d-Hermite-0::0.00000000 0.00000000 1.00000000 0 0 0 1.00000000 1.00000000 1.00000000 1.00000000 2.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-3d-Hermite-0::
+DEAL:Cell-3d-Hermite-0::1.00000000 0.00000000 1.00000000 0 0 0 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 2.00000000 1.00000000 1.00000000
+DEAL:Cell-3d-Hermite-0::
+DEAL:Cell-3d-Hermite-0::
+DEAL:Cell-3d-Hermite-0::0.00000000 1.00000000 1.00000000 0 0 0 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 2.00000000 1.00000000
+DEAL:Cell-3d-Hermite-0::
+DEAL:Cell-3d-Hermite-0::1.00000000 1.00000000 1.00000000 0 0 0 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 2.00000000
+DEAL:Cell-3d-Hermite-0::
+DEAL:Cell-3d-Hermite-0::
+DEAL:Cell-3d-Hermite-0::
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+DEAL:Cell-3d-Hermite-1::
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+DEAL:Cell-3d-Hermite-1::1.00000000 0.00000000 0.00000000 1 1 1 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 65.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-3d-Hermite-1::
+DEAL:Cell-3d-Hermite-1::
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+DEAL:Cell-3d-Hermite-1::0.00000000 1.00000000 0.00000000 0 1 0 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 5.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
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+DEAL:Cell-3d-Hermite-1::0.00000000 1.00000000 0.00000000 1 0 1 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 17.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-3d-Hermite-1::0.00000000 1.00000000 0.00000000 0 1 1 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 17.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-3d-Hermite-1::0.00000000 1.00000000 0.00000000 1 1 1 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 65.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-3d-Hermite-1::
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+DEAL:Cell-3d-Hermite-1::
+DEAL:Cell-3d-Hermite-1::
+DEAL:Cell-3d-Hermite-1::
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+DEAL:Cell-3d-Hermite-1::
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+DEAL:Cell-3d-Hermite-1::1.00000000 0.00000000 1.00000000 0 1 1 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 17.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-3d-Hermite-1::1.00000000 0.00000000 1.00000000 1 1 1 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 65.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-3d-Hermite-1::
+DEAL:Cell-3d-Hermite-1::
+DEAL:Cell-3d-Hermite-1::0.00000000 1.00000000 1.00000000 0 0 0 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 2.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-3d-Hermite-1::0.00000000 1.00000000 1.00000000 1 0 0 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 5.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-3d-Hermite-1::0.00000000 1.00000000 1.00000000 0 1 0 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 5.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-3d-Hermite-1::0.00000000 1.00000000 1.00000000 1 1 0 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 17.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-3d-Hermite-1::0.00000000 1.00000000 1.00000000 0 0 1 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 5.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-3d-Hermite-1::0.00000000 1.00000000 1.00000000 1 0 1 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 17.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-3d-Hermite-1::0.00000000 1.00000000 1.00000000 0 1 1 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 17.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-3d-Hermite-1::0.00000000 1.00000000 1.00000000 1 1 1 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 65.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-3d-Hermite-1::
+DEAL:Cell-3d-Hermite-1::1.00000000 1.00000000 1.00000000 0 0 0 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 2.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-3d-Hermite-1::1.00000000 1.00000000 1.00000000 1 0 0 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 5.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-3d-Hermite-1::1.00000000 1.00000000 1.00000000 0 1 0 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 5.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-3d-Hermite-1::1.00000000 1.00000000 1.00000000 1 1 0 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 17.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-3d-Hermite-1::1.00000000 1.00000000 1.00000000 0 0 1 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 5.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-3d-Hermite-1::1.00000000 1.00000000 1.00000000 1 0 1 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 17.00000000 1.00000000 1.00000000
+DEAL:Cell-3d-Hermite-1::1.00000000 1.00000000 1.00000000 0 1 1 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 17.00000000 1.00000000
+DEAL:Cell-3d-Hermite-1::1.00000000 1.00000000 1.00000000 1 1 1 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 65.00000000
+DEAL:Cell-3d-Hermite-1::
+DEAL:Cell-3d-Hermite-1::
+DEAL:Cell-3d-Hermite-1::
//
// ---------------------------------------------------------------------
+#ifndef SHAPES_TEST_HEADER
+#define SHAPES_TEST_HEADER
// Show the shape functions implemented.
check_values_and_derivatives(fe, fe_values, q);
};
}
+
+#endif
--- /dev/null
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2013 - 2022 by the deal.II authors
+//
+// This file is part of the deal.II library.
+//
+// The deal.II library is free software; you can use it, redistribute
+// it, and/or modify it under the terms of the GNU Lesser General
+// Public License as published by the Free Software Foundation; either
+// version 2.1 of the License, or (at your option) any later version.
+// The full text of the license can be found in the file LICENSE.md at
+// the top level directory of deal.II.
+//
+// ---------------------------------------------------------------------
+
+#include <deal.II/fe/fe_hermite.h>
+#include <deal.II/fe/mapping_cartesian.h>
+
+#include <string>
+
+#include "../tests.h"
+
+#include "shapes.h"
+
+#define PRECISION 8
+
+/*
+ * Test file to check that the shape values of an Hermite polynomial basis
+ * of a given regularity are correct. Values of derivatives at the boundaries
+ * are checked elsehwere in derivatives_hermite.cc.
+ */
+
+template <int dim>
+void
+plot_FE_Hermite_shape_functions()
+{
+ MappingCartesian<dim> m;
+
+ FE_Hermite<dim> herm0(0);
+ plot_shape_functions(m, herm0, "Hermite-0");
+ plot_face_shape_functions(m, herm0, "Hermite-0");
+ test_compute_functions(m, herm0, "Hermite-0");
+
+ FE_Hermite<dim> herm1(1);
+ plot_shape_functions(m, herm1, "Hermite-1");
+ plot_face_shape_functions(m, herm1, "Hermite-1");
+ test_compute_functions(m, herm1, "Hermite-1");
+
+ // skip the following tests to
+ // reduce run-time
+ if (dim < 3)
+ {
+ FE_Hermite<dim> herm2(2);
+ plot_shape_functions(m, herm2, "Hermite-2");
+ plot_face_shape_functions(m, herm2, "Hermite-2");
+ test_compute_functions(m, herm2, "Hermite-2");
+ }
+
+ if (dim == 1)
+ {
+ FE_Hermite<dim> herm3(3);
+ plot_shape_functions(m, herm3, "Hermite-3");
+ plot_face_shape_functions(m, herm3, "Hermite-3");
+ test_compute_functions(m, herm3, "Hermite-3");
+
+ FE_Hermite<dim> herm4(4);
+ plot_shape_functions(m, herm4, "Hermite-4");
+ plot_face_shape_functions(m, herm4, "Hermite-4");
+ test_compute_functions(m, herm4, "Hermite-4");
+ };
+}
+
+
+
+int
+main()
+{
+ std::ofstream logfile("output");
+ deallog << std::setprecision(PRECISION) << std::fixed;
+ deallog.attach(logfile);
+
+ plot_FE_Hermite_shape_functions<1>();
+ plot_FE_Hermite_shape_functions<2>();
+ plot_FE_Hermite_shape_functions<3>();
+
+ return 0;
+}
--- /dev/null
+
+DEAL:Cell1d-Hermite-0::0.00000000 2.00000000 1.00000000
+DEAL:Cell1d-Hermite-0::0.25000000 1.75000000 1.25000000
+DEAL:Cell1d-Hermite-0::0.50000000 1.50000000 1.50000000
+DEAL:Cell1d-Hermite-0::0.75000000 1.25000000 1.75000000
+DEAL:Cell1d-Hermite-0::1.00000000 1.00000000 2.00000000
+DEAL:Cell1d-Hermite-0::
+DEAL:Cell1d-Hermite-0::
+DEAL:Cell1d-Hermite-1::0.00000000 2.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell1d-Hermite-1::0.25000000 1.84375000 1.56250000 1.15625000 0.81250000
+DEAL:Cell1d-Hermite-1::0.50000000 1.50000000 1.50000000 1.50000000 0.50000000
+DEAL:Cell1d-Hermite-1::0.75000000 1.15625000 1.18750000 1.84375000 0.43750000
+DEAL:Cell1d-Hermite-1::1.00000000 1.00000000 1.00000000 2.00000000 1.00000000
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+DEAL:Cell2d-Hermite-2::
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+DEAL:Face2d-Hermite-2::
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+DEAL:Face2d-Hermite-2::
+DEAL:Face2d-Hermite-2::
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+DEAL:Face2d-Hermite-2::
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+DEAL:Face2d-Hermite-2::
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+DEAL:Face2d-Hermite-2::
+DEAL:Face2d-Hermite-2::
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+DEAL:Cell3d-Hermite-0::0.50000000 0.00000000 0.00000000 1.50000000 1.50000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell3d-Hermite-0::0.75000000 0.00000000 0.00000000 1.25000000 1.75000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell3d-Hermite-0::1.00000000 0.00000000 0.00000000 1.00000000 2.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell3d-Hermite-0::
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+DEAL:Face3d-Hermite-1::
+DEAL:Face3d-Hermite-1::0.00000000 0.37500000 0.50000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.15625000 1.00000000 1.09375000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.84375000 1.00000000 0.71875000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Face3d-Hermite-1::0.12500000 0.37500000 0.50000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.13183594 1.04394531 1.07910156 1.02636719 1.00000000 1.00000000 1.00000000 1.00000000 1.02441406 0.98535156 1.01464844 0.99121094 1.00000000 1.00000000 1.00000000 1.00000000 1.71191406 1.23730469 0.76269531 0.92089844 1.00000000 1.00000000 1.00000000 1.00000000 1.13183594 0.92089844 0.95605469 1.02636719 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Face3d-Hermite-1::0.25000000 0.37500000 0.50000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.07812500 1.03906250 1.04687500 1.02343750 1.00000000 1.00000000 1.00000000 1.00000000 1.07812500 0.96093750 1.04687500 0.97656250 1.00000000 1.00000000 1.00000000 1.00000000 1.42187500 1.21093750 0.85937500 0.92968750 1.00000000 1.00000000 1.00000000 1.00000000 1.42187500 0.78906250 0.85937500 1.07031250 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Face3d-Hermite-1::0.37500000 0.37500000 0.50000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.02441406 1.01464844 1.01464844 1.00878906 1.00000000 1.00000000 1.00000000 1.00000000 1.13183594 0.95605469 1.07910156 0.97363281 1.00000000 1.00000000 1.00000000 1.00000000 1.13183594 1.07910156 0.95605469 0.97363281 1.00000000 1.00000000 1.00000000 1.00000000 1.71191406 0.76269531 0.76269531 1.07910156 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Face3d-Hermite-1::0.50000000 0.37500000 0.50000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.15625000 1.00000000 1.09375000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.84375000 1.00000000 0.71875000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Face3d-Hermite-1::
+DEAL:Face3d-Hermite-1::0.00000000 0.50000000 0.50000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 2.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Face3d-Hermite-1::0.12500000 0.50000000 0.50000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.84375000 1.28125000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.15625000 0.90625000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Face3d-Hermite-1::0.25000000 0.50000000 0.50000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.50000000 1.25000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.50000000 0.75000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Face3d-Hermite-1::0.37500000 0.50000000 0.50000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.15625000 1.09375000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.84375000 0.71875000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Face3d-Hermite-1::0.50000000 0.50000000 0.50000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 2.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Face3d-Hermite-1::
+DEAL:Face3d-Hermite-1::
--- /dev/null
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2021 - 2022 by the deal.II authors
+//
+// This file is part of the deal.II library.
+//
+// The deal.II library is free software; you can use it, redistribute
+// it, and/or modify it under the terms of the GNU Lesser General
+// Public License as published by the Free Software Foundation; either
+// version 2.1 of the License, or (at your option) any later version.
+// The full text of the license can be found in the file LICENSE.md at
+// the top level directory of deal.II.
+//
+// ---------------------------------------------------------------------
+
+#include <deal.II/base/function.h>
+#include <deal.II/base/quadrature_lib.h>
+#include <deal.II/base/utilities.h>
+
+#include <deal.II/dofs/dof_accessor.h>
+#include <deal.II/dofs/dof_handler.h>
+#include <deal.II/dofs/dof_tools.h>
+
+#include <deal.II/fe/fe_hermite.h>
+#include <deal.II/fe/fe_interface_values.h>
+#include <deal.II/fe/fe_values.h>
+#include <deal.II/fe/mapping_cartesian.h>
+
+#include <deal.II/grid/grid_generator.h>
+#include <deal.II/grid/tria.h>
+#include <deal.II/grid/tria_accessor.h>
+#include <deal.II/grid/tria_iterator.h>
+
+#include <deal.II/lac/affine_constraints.h>
+#include <deal.II/lac/dynamic_sparsity_pattern.h>
+#include <deal.II/lac/full_matrix.h>
+#include <deal.II/lac/precondition.h>
+#include <deal.II/lac/solver_cg.h>
+#include <deal.II/lac/sparse_matrix.h>
+#include <deal.II/lac/vector.h>
+
+#include <deal.II/numerics/data_out.h>
+#include <deal.II/numerics/matrix_tools.h>
+#include <deal.II/numerics/vector_tools.h>
+
+#include <cmath>
+#include <cstdlib>
+#include <fstream>
+#include <iterator>
+#include <sstream>
+#include <vector>
+
+
+
+/*
+ * Test case for Hermite on an irregular 1D grid.
+ * <code>FE_Hermite<1>(reg)<\code> should be able to perfectly represent any
+ * polynomial function up to degree $2 \times reg+1$. If all basis functions
+ * are correctly scaled according to element size, then projecting a
+ * polynomial of this form onto the Hermite FE space will produce negligible
+ * pointwise errors.
+ */
+using namespace dealii;
+
+// Define a function to project onto the domain
+class TestPoly : public Function<1>
+{
+public:
+ virtual double
+ value(const Point<1> &p, unsigned int c = 0) const override
+ {
+ return p(c) * (1.0 + p(c) * (0.5 - p(c)));
+ }
+
+
+
+ std::string
+ get_polynomial_string()
+ {
+ return "X + 0.5 X^2 - X^3";
+ }
+};
+
+
+
+void
+test_fe_on_domain(const unsigned int regularity)
+{
+ Triangulation<1> tr;
+ DoFHandler<1> dof(tr);
+
+ double left = -1.0, right = 1.0;
+ Point<1> left_point(left), right_point(right);
+ GridGenerator::hyper_cube(tr, left, right);
+
+ // Refine the right-most cell three times to get the elements
+ // [-1,0],[0,0.5],[0.5,0.75],[0.75,1]
+ for (unsigned int i = 0; i < 3; ++i)
+ {
+ for (auto &cell : tr.active_cell_iterators())
+ {
+ const double distance = right_point.distance(cell->vertex(1));
+ if (distance < 1e-6)
+ {
+ cell->set_refine_flag();
+ // break;
+ }
+ }
+ tr.execute_coarsening_and_refinement();
+ }
+
+ FE_Hermite<1> herm(regularity);
+ dof.distribute_dofs(herm);
+
+ MappingCartesian<1> mapping;
+
+ QGauss<1> quadr(2 * regularity + 2);
+ Vector<double> solution(dof.n_dofs());
+ TestPoly rhs_func;
+
+ AffineConstraints<double> constraints;
+ constraints.close();
+
+ FEValues<1> fe_herm(mapping,
+ herm,
+ quadr,
+ update_values | update_quadrature_points |
+ update_JxW_values);
+
+ std::vector<types::global_dof_index> local_to_global(herm.n_dofs_per_cell());
+
+ VectorTools::project_hermite(
+ mapping, dof, constraints, quadr, rhs_func, solution, false);
+
+ double err_sq = 0;
+
+ for (auto &cell : dof.active_cell_iterators())
+ {
+ fe_herm.reinit(cell);
+ cell->get_dof_indices(local_to_global);
+ for (const unsigned int q : fe_herm.quadrature_point_indices())
+ {
+ double sol_at_point = 0;
+ for (const unsigned int i : fe_herm.dof_indices())
+ sol_at_point +=
+ fe_herm.shape_value(i, q) * solution(local_to_global[i]);
+
+ sol_at_point -= rhs_func.value(fe_herm.quadrature_point(q));
+ err_sq += sol_at_point * sol_at_point * fe_herm.JxW(q);
+ }
+ }
+
+ err_sq = std::sqrt(err_sq);
+
+ char fname[50];
+ sprintf(fname, "Cell-1d-Hermite-%d", regularity);
+ deallog.push(fname);
+
+ deallog << "Test polynomial:" << std::endl;
+ deallog << rhs_func.get_polynomial_string() << std::endl;
+ deallog << std::endl;
+
+ deallog << "Grid cells:" << std::endl;
+ for (const auto &cell : tr.active_cell_iterators())
+ {
+ deallog << "(\t" << cell->vertex(0) << ","
+ << "\t" << cell->vertex(1) << "\t)" << std::endl;
+ }
+ deallog << std::endl;
+
+ deallog << "Interpolation error:" << std::endl;
+ deallog << err_sq << "\n\n" << std::endl;
+ deallog.pop();
+}
+
+
+
+int
+main()
+{
+ std::ofstream logfile("output");
+
+ deallog << std::setprecision(8) << std::fixed;
+ deallog.attach(logfile);
+
+ test_fe_on_domain(0);
+ test_fe_on_domain(1);
+ test_fe_on_domain(2);
+ test_fe_on_domain(3);
+
+ return 0;
+}
--- /dev/null
+
+DEAL:Cell-1d-Hermite-0::Test polynomial:
+DEAL:Cell-1d-Hermite-0::X + 0.5 X^2 - X^3
+DEAL:Cell-1d-Hermite-0::
+DEAL:Cell-1d-Hermite-0::Grid cells:
+DEAL:Cell-1d-Hermite-0::( -1.00000000, 0.00000000 )
+DEAL:Cell-1d-Hermite-0::( 0.00000000, 0.50000000 )
+DEAL:Cell-1d-Hermite-0::( 0.50000000, 0.75000000 )
+DEAL:Cell-1d-Hermite-0::( 0.75000000, 1.00000000 )
+DEAL:Cell-1d-Hermite-0::
+DEAL:Cell-1d-Hermite-0::Interpolation error:
+DEAL:Cell-1d-Hermite-0::0.07498485
+
+
+DEAL:Cell-1d-Hermite-1::Test polynomial:
+DEAL:Cell-1d-Hermite-1::X + 0.5 X^2 - X^3
+DEAL:Cell-1d-Hermite-1::
+DEAL:Cell-1d-Hermite-1::Grid cells:
+DEAL:Cell-1d-Hermite-1::( -1.00000000, 0.00000000 )
+DEAL:Cell-1d-Hermite-1::( 0.00000000, 0.50000000 )
+DEAL:Cell-1d-Hermite-1::( 0.50000000, 0.75000000 )
+DEAL:Cell-1d-Hermite-1::( 0.75000000, 1.00000000 )
+DEAL:Cell-1d-Hermite-1::
+DEAL:Cell-1d-Hermite-1::Interpolation error:
+DEAL:Cell-1d-Hermite-1::0.00000000
+
+
+DEAL:Cell-1d-Hermite-2::Test polynomial:
+DEAL:Cell-1d-Hermite-2::X + 0.5 X^2 - X^3
+DEAL:Cell-1d-Hermite-2::
+DEAL:Cell-1d-Hermite-2::Grid cells:
+DEAL:Cell-1d-Hermite-2::( -1.00000000, 0.00000000 )
+DEAL:Cell-1d-Hermite-2::( 0.00000000, 0.50000000 )
+DEAL:Cell-1d-Hermite-2::( 0.50000000, 0.75000000 )
+DEAL:Cell-1d-Hermite-2::( 0.75000000, 1.00000000 )
+DEAL:Cell-1d-Hermite-2::
+DEAL:Cell-1d-Hermite-2::Interpolation error:
+DEAL:Cell-1d-Hermite-2::0.00000000
+
+
+DEAL:Cell-1d-Hermite-3::Test polynomial:
+DEAL:Cell-1d-Hermite-3::X + 0.5 X^2 - X^3
+DEAL:Cell-1d-Hermite-3::
+DEAL:Cell-1d-Hermite-3::Grid cells:
+DEAL:Cell-1d-Hermite-3::( -1.00000000, 0.00000000 )
+DEAL:Cell-1d-Hermite-3::( 0.00000000, 0.50000000 )
+DEAL:Cell-1d-Hermite-3::( 0.50000000, 0.75000000 )
+DEAL:Cell-1d-Hermite-3::( 0.75000000, 1.00000000 )
+DEAL:Cell-1d-Hermite-3::
+DEAL:Cell-1d-Hermite-3::Interpolation error:
+DEAL:Cell-1d-Hermite-3::0.00000000
+
+
--- /dev/null
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2021 - 2022 by the deal.II authors
+//
+// This file is part of the deal.II library.
+//
+// The deal.II library is free software; you can use it, redistribute
+// it, and/or modify it under the terms of the GNU Lesser General
+// Public License as published by the Free Software Foundation; either
+// version 2.1 of the License, or (at your option) any later version.
+// The full text of the license can be found in the file LICENSE.md at
+// the top level directory of deal.II.
+//
+// ---------------------------------------------------------------------
+
+#include <deal.II/base/function.h>
+#include <deal.II/base/quadrature_lib.h>
+#include <deal.II/base/utilities.h>
+
+#include <deal.II/dofs/dof_accessor.h>
+#include <deal.II/dofs/dof_handler.h>
+#include <deal.II/dofs/dof_tools.h>
+
+#include <deal.II/fe/fe_hermite.h>
+#include <deal.II/fe/fe_interface_values.h>
+#include <deal.II/fe/fe_values.h>
+#include <deal.II/fe/mapping_cartesian.h>
+
+#include <deal.II/grid/grid_generator.h>
+#include <deal.II/grid/tria.h>
+#include <deal.II/grid/tria_accessor.h>
+#include <deal.II/grid/tria_iterator.h>
+
+#include <deal.II/lac/affine_constraints.h>
+#include <deal.II/lac/dynamic_sparsity_pattern.h>
+#include <deal.II/lac/full_matrix.h>
+#include <deal.II/lac/precondition.h>
+#include <deal.II/lac/solver_cg.h>
+#include <deal.II/lac/sparse_matrix.h>
+#include <deal.II/lac/vector.h>
+
+#include <deal.II/numerics/data_out.h>
+#include <deal.II/numerics/matrix_tools.h>
+
+#include <cmath>
+#include <cstdlib>
+#include <fstream>
+#include <iterator>
+#include <sstream>
+#include <vector>
+
+
+
+/*
+ * Test case for Hermite on an irregular 1D grid.
+ * <code>FE_Hermite<1>(reg)<\code> should be able to perfectly represent any
+ * polynomial function up to degree $2 \times reg+1$. If all basis functions
+ * are correctly scaled according to element size, then solving the Laplace
+ * equation for a polynomial solution of this form in the Hermite FE space
+ * will produce negligible pointwise errors.
+ */
+
+using namespace dealii;
+
+// Define a function to project onto the domain [-1,1]
+class Solution : public Function<1>
+{
+public:
+ virtual double
+ value(const Point<1> &p, unsigned int c = 0) const override
+ {
+ return p(c) * (1.0 + p(c) * (0.5 - p(c)));
+ }
+
+
+
+ std::string
+ get_function_string()
+ {
+ return "X + 0.5 X^2 - X^3";
+ }
+};
+
+
+
+class RHSFunction : public Function<1>
+{
+public:
+ virtual double
+ value(const Point<1> &p, unsigned int c = 0) const override
+ {
+ return -1.0 + 6 * p(c);
+ }
+};
+
+
+
+void
+test_fe_on_domain(const unsigned int regularity)
+{
+ deallog << std::endl;
+ char fname[50];
+ sprintf(fname, "Cell-1d-Hermite-%d", regularity);
+ deallog.push(fname);
+ deallog.depth_file(2);
+
+ Triangulation<1> tr;
+ DoFHandler<1> dof(tr);
+
+ double left = -1.0, right = 1.0;
+ Point<1> left_point(left), right_point(right);
+ GridGenerator::hyper_cube(tr, left, right);
+
+ // Refine the right-most cell three times to get the elements
+ // [-1,0],[0,0.5],[0.5,0.75],[0.75,1]
+ for (unsigned int i = 0; i < 3; ++i)
+ {
+ for (auto &cell : tr.active_cell_iterators())
+ {
+ const double distance = right_point.distance(cell->vertex(1));
+ if (distance < 1e-6)
+ {
+ cell->set_refine_flag();
+ }
+ }
+ tr.execute_coarsening_and_refinement();
+ }
+
+ FE_Hermite<1> herm(regularity);
+ dof.distribute_dofs(herm);
+
+ MappingCartesian<1> mapping;
+
+ QGauss<1> quadr(2 * regularity + 2);
+
+ Vector<double> sol(dof.n_dofs());
+ Vector<double> rhs(dof.n_dofs());
+
+ Solution sol_object;
+ RHSFunction rhs_object;
+
+ AffineConstraints<double> constraints;
+ constraints.close();
+
+ DynamicSparsityPattern dsp(dof.n_dofs());
+ DoFTools::make_sparsity_pattern(dof, dsp);
+ SparsityPattern sp;
+ sp.copy_from(dsp);
+
+ SparseMatrix<double> stiffness_matrix;
+ stiffness_matrix.reinit(sp);
+
+ MatrixCreator::create_laplace_matrix(mapping, dof, quadr, stiffness_matrix);
+
+ FEValues<1> fe_herm(mapping,
+ herm,
+ quadr,
+ update_values | update_gradients |
+ update_quadrature_points | update_JxW_values);
+
+ std::vector<types::global_dof_index> local_to_global(herm.n_dofs_per_cell());
+
+ for (const auto &cell : dof.active_cell_iterators())
+ {
+ fe_herm.reinit(cell);
+ cell->get_dof_indices(local_to_global);
+ for (const unsigned int i : fe_herm.dof_indices())
+ {
+ double rhs_temp = 0;
+ for (const unsigned int q : fe_herm.quadrature_point_indices())
+ rhs_temp += fe_herm.shape_value(i, q) *
+ rhs_object.value(fe_herm.quadrature_point(q)) *
+ fe_herm.JxW(q);
+ rhs(local_to_global[i]) += rhs_temp;
+ }
+ }
+
+ std::map<types::global_dof_index, double> bound_vals;
+
+ std::map<types::boundary_id, const Function<1, double> *> bound_map;
+ bound_map.emplace(std::make_pair(0U, &sol_object));
+ bound_map.emplace(std::make_pair(1U, &sol_object));
+
+ VectorTools::project_hermite_boundary_values(
+ mapping,
+ dof,
+ bound_map,
+ QGauss<0>(1),
+ VectorTools::HermiteBoundaryType::hermite_dirichlet,
+ bound_vals);
+
+ MatrixTools::apply_boundary_values(bound_vals, stiffness_matrix, sol, rhs);
+
+ IterationNumberControl solver_control_values(50, 1e-11);
+ SolverCG<> solver(solver_control_values);
+
+ solver.solve(stiffness_matrix, sol, rhs, PreconditionIdentity());
+
+ double err_sq = 0;
+
+ for (auto &cell : dof.active_cell_iterators())
+ {
+ fe_herm.reinit(cell);
+ cell->get_dof_indices(local_to_global);
+ for (const unsigned int q : fe_herm.quadrature_point_indices())
+ {
+ double sol_at_point = 0;
+ for (const unsigned int i : fe_herm.dof_indices())
+ sol_at_point += fe_herm.shape_value(i, q) * sol(local_to_global[i]);
+
+ sol_at_point -= sol_object.value(fe_herm.quadrature_point(q));
+ err_sq += sol_at_point * sol_at_point * fe_herm.JxW(q);
+ }
+ }
+
+ err_sq = std::sqrt(err_sq);
+
+ deallog << "Test polynomial:" << std::endl;
+ deallog << sol_object.get_function_string() << std::endl;
+ deallog << std::endl;
+
+ deallog << "Grid cells:" << std::endl;
+ for (const auto &cell : tr.active_cell_iterators())
+ {
+ deallog << "(\t" << cell->vertex(0) << ","
+ << "\t" << cell->vertex(1) << "\t)" << std::endl;
+ }
+ deallog << std::endl;
+
+ deallog << "Interpolation error:" << std::endl;
+ deallog << err_sq << "\n\n" << std::endl;
+ deallog.pop();
+}
+
+
+
+int
+main()
+{
+ std::ofstream logfile("output");
+ deallog << std::setprecision(8) << std::fixed;
+ deallog.attach(logfile);
+
+ test_fe_on_domain(0);
+ test_fe_on_domain(1);
+ test_fe_on_domain(2);
+ test_fe_on_domain(3);
+
+ return 0;
+}
--- /dev/null
+
+DEAL::
+DEAL:Cell-1d-Hermite-0::Test polynomial:
+DEAL:Cell-1d-Hermite-0::X + 0.5 X^2 - X^3
+DEAL:Cell-1d-Hermite-0::
+DEAL:Cell-1d-Hermite-0::Grid cells:
+DEAL:Cell-1d-Hermite-0::( -1.00000000, 0.00000000 )
+DEAL:Cell-1d-Hermite-0::( 0.00000000, 0.50000000 )
+DEAL:Cell-1d-Hermite-0::( 0.50000000, 0.75000000 )
+DEAL:Cell-1d-Hermite-0::( 0.75000000, 1.00000000 )
+DEAL:Cell-1d-Hermite-0::
+DEAL:Cell-1d-Hermite-0::Interpolation error:
+DEAL:Cell-1d-Hermite-0::0.33715327
+
+
+DEAL::
+DEAL:Cell-1d-Hermite-1::Test polynomial:
+DEAL:Cell-1d-Hermite-1::X + 0.5 X^2 - X^3
+DEAL:Cell-1d-Hermite-1::
+DEAL:Cell-1d-Hermite-1::Grid cells:
+DEAL:Cell-1d-Hermite-1::( -1.00000000, 0.00000000 )
+DEAL:Cell-1d-Hermite-1::( 0.00000000, 0.50000000 )
+DEAL:Cell-1d-Hermite-1::( 0.50000000, 0.75000000 )
+DEAL:Cell-1d-Hermite-1::( 0.75000000, 1.00000000 )
+DEAL:Cell-1d-Hermite-1::
+DEAL:Cell-1d-Hermite-1::Interpolation error:
+DEAL:Cell-1d-Hermite-1::0.00000000
+
+
+DEAL::
+DEAL:Cell-1d-Hermite-2::Test polynomial:
+DEAL:Cell-1d-Hermite-2::X + 0.5 X^2 - X^3
+DEAL:Cell-1d-Hermite-2::
+DEAL:Cell-1d-Hermite-2::Grid cells:
+DEAL:Cell-1d-Hermite-2::( -1.00000000, 0.00000000 )
+DEAL:Cell-1d-Hermite-2::( 0.00000000, 0.50000000 )
+DEAL:Cell-1d-Hermite-2::( 0.50000000, 0.75000000 )
+DEAL:Cell-1d-Hermite-2::( 0.75000000, 1.00000000 )
+DEAL:Cell-1d-Hermite-2::
+DEAL:Cell-1d-Hermite-2::Interpolation error:
+DEAL:Cell-1d-Hermite-2::0.00000000
+
+
+DEAL::
+DEAL:Cell-1d-Hermite-3::Test polynomial:
+DEAL:Cell-1d-Hermite-3::X + 0.5 X^2 - X^3
+DEAL:Cell-1d-Hermite-3::
+DEAL:Cell-1d-Hermite-3::Grid cells:
+DEAL:Cell-1d-Hermite-3::( -1.00000000, 0.00000000 )
+DEAL:Cell-1d-Hermite-3::( 0.00000000, 0.50000000 )
+DEAL:Cell-1d-Hermite-3::( 0.50000000, 0.75000000 )
+DEAL:Cell-1d-Hermite-3::( 0.75000000, 1.00000000 )
+DEAL:Cell-1d-Hermite-3::
+DEAL:Cell-1d-Hermite-3::Interpolation error:
+DEAL:Cell-1d-Hermite-3::0.00000000
+
+
--- /dev/null
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2021 - 2022 by the deal.II authors
+//
+// This file is part of the deal.II library.
+//
+// The deal.II library is free software; you can use it, redistribute
+// it, and/or modify it under the terms of the GNU Lesser General
+// Public License as published by the Free Software Foundation; either
+// version 2.1 of the License, or (at your option) any later version.
+// The full text of the license can be found in the file LICENSE.md at
+// the top level directory of deal.II.
+//
+// ---------------------------------------------------------------------
+
+#include <deal.II/base/config.h>
+
+#include <deal.II/base/function.h>
+#include <deal.II/base/quadrature_lib.h>
+#include <deal.II/base/utilities.h>
+
+#include <deal.II/dofs/dof_accessor.h>
+#include <deal.II/dofs/dof_handler.h>
+#include <deal.II/dofs/dof_tools.h>
+
+#include <deal.II/fe/fe_hermite.h>
+#include <deal.II/fe/fe_interface_values.h>
+#include <deal.II/fe/fe_values.h>
+#include <deal.II/fe/mapping_cartesian.h>
+
+#include <deal.II/grid/grid_generator.h>
+#include <deal.II/grid/tria.h>
+#include <deal.II/grid/tria_accessor.h>
+#include <deal.II/grid/tria_iterator.h>
+
+#include <deal.II/lac/affine_constraints.h>
+#include <deal.II/lac/dynamic_sparsity_pattern.h>
+#include <deal.II/lac/full_matrix.h>
+#include <deal.II/lac/precondition.h>
+#include <deal.II/lac/solver_cg.h>
+#include <deal.II/lac/sparse_matrix.h>
+#include <deal.II/lac/vector.h>
+
+#include <deal.II/numerics/data_out.h>
+#include <deal.II/numerics/matrix_tools.h>
+#include <deal.II/numerics/vector_tools.h>
+
+#include <cmath>
+#include <cstdlib>
+#include <fstream>
+#include <iterator>
+#include <sstream>
+#include <vector>
+
+
+
+/*
+ * Test case for Hermite on an irregular 1D grid.
+ * <code>FE_Hermite<dim>(reg)<\code> should be able to perfectly represent any
+ * polynomial function up to degree $2 \times reg+1$, including on the
+ * boundaries. If all basis functions are correctly scaled according to
+ * element size, then solving the Laplace equation with a polynomial solution
+ * in the Hermite FE space will produce negligible pointwise errors for
+ * non-homogeneous Dirichlet boundary conditions.
+ */
+using namespace dealii;
+
+// Define a function to project onto the domain [-1,1]^dim
+template <int dim>
+class Solution : public Function<dim>
+{
+public:
+ virtual double
+ value(const Point<dim> &p, unsigned int c = 0) const override
+ {
+ (void)c;
+ if (dim == 0)
+ Assert(false, ExcNotImplemented());
+ double temp = p(0) * (1.0 + p(0) * (0.5 - p(0)));
+ if (dim > 1)
+ temp *= 1.0 - p(1) * p(1);
+ if (dim == 3)
+ temp *= p(2);
+ return temp;
+ }
+
+
+
+ std::string
+ get_function_string()
+ {
+ switch (dim)
+ {
+ case 1:
+ return "X + 0.5 X^2 - X^3";
+ case 2:
+ return "(X + 0.5 X^2 - X^3)(1 - Y^2)";
+ case 3:
+ return "(X + 0.5 X^2 - X^3)(1 - Y^2)Z";
+ default:
+ Assert(false, ExcNotImplemented());
+ return "";
+ }
+ }
+};
+
+
+
+// This function should be the analytical Laplacian of the solution
+// function above.
+template <int dim>
+class RHSFunction : public Function<dim>
+{
+public:
+ virtual double
+ value(const Point<dim> &p, unsigned int c = 0) const override
+ {
+ (void)c;
+ if (dim == 0)
+ Assert(false, ExcNotImplemented());
+ double temp = -1.0 + 6 * p(0);
+ if (dim > 1)
+ {
+ temp *= -p(1) * p(1);
+ temp -= 1.0 - 8.0 * p(0) - p(0) * p(0) + 2.0 * p(0) * p(0) * p(0);
+ }
+ if (dim == 3)
+ temp *= p(2);
+ return temp;
+ }
+};
+
+
+
+template <int dim>
+void
+test_fe_on_domain(const unsigned int regularity)
+{
+ deallog << std::endl;
+ char fname[50];
+ sprintf(fname, "Cell-%dd-Hermite-%d", dim, regularity);
+ deallog.push(fname);
+ deallog.depth_file(2);
+
+ Triangulation<dim> tr;
+ DoFHandler<dim> dof(tr);
+
+ double left = -1.0, right = 1.0;
+ GridGenerator::subdivided_hyper_cube(tr, 4, left, right);
+
+ FE_Hermite<dim> herm(regularity);
+ dof.distribute_dofs(herm);
+
+ MappingCartesian<dim> mapping;
+
+ QGauss<dim> quadr(2 * regularity + 2);
+
+ Vector<double> sol(dof.n_dofs());
+ Vector<double> rhs(dof.n_dofs());
+
+ Solution<dim> sol_object;
+ RHSFunction<dim> rhs_object;
+
+ AffineConstraints<double> constraints;
+ constraints.close();
+
+ DynamicSparsityPattern dsp(dof.n_dofs());
+ DoFTools::make_sparsity_pattern(dof, dsp);
+ SparsityPattern sp;
+ sp.copy_from(dsp);
+
+ SparseMatrix<double> stiffness_matrix;
+ stiffness_matrix.reinit(sp);
+ MatrixCreator::create_laplace_matrix(mapping, dof, quadr, stiffness_matrix);
+
+ FEValues<dim> fe_herm(mapping,
+ herm,
+ quadr,
+ update_values | update_gradients |
+ update_quadrature_points | update_JxW_values);
+
+ std::vector<types::global_dof_index> local_to_global(herm.n_dofs_per_cell());
+
+ for (const auto &cell : dof.active_cell_iterators())
+ {
+ fe_herm.reinit(cell);
+ cell->get_dof_indices(local_to_global);
+ for (const unsigned int i : fe_herm.dof_indices())
+ {
+ double rhs_temp = 0;
+ for (const unsigned int q : fe_herm.quadrature_point_indices())
+ rhs_temp += fe_herm.shape_value(i, q) *
+ rhs_object.value(fe_herm.quadrature_point(q)) *
+ fe_herm.JxW(q);
+
+ rhs(local_to_global[i]) += rhs_temp;
+ }
+ }
+
+ std::map<types::global_dof_index, double> bound_vals;
+
+ std::map<types::boundary_id, const Function<dim, double> *> bound_map;
+ bound_map.emplace(std::make_pair(0U, &sol_object));
+ if (dim == 1)
+ bound_map.emplace(std::make_pair(1U, &sol_object));
+
+ // The following is the main function being tested here
+ VectorTools::project_hermite_boundary_values(
+ mapping,
+ dof,
+ bound_map,
+ QGauss<dim - 1>(2 * regularity + 2),
+ VectorTools::HermiteBoundaryType::hermite_dirichlet,
+ bound_vals);
+
+ MatrixTools::apply_boundary_values(bound_vals, stiffness_matrix, sol, rhs);
+
+ IterationNumberControl solver_control_values(8000, 1e-11);
+ SolverCG<> solver(solver_control_values);
+
+ solver.solve(stiffness_matrix, sol, rhs, PreconditionIdentity());
+
+ double err_sq = 0;
+
+ for (auto &cell : dof.active_cell_iterators())
+ {
+ fe_herm.reinit(cell);
+ cell->get_dof_indices(local_to_global);
+ for (const unsigned int q : fe_herm.quadrature_point_indices())
+ {
+ double sol_at_point = 0;
+ for (const unsigned int i : fe_herm.dof_indices())
+ sol_at_point += fe_herm.shape_value(i, q) * sol(local_to_global[i]);
+
+ sol_at_point -= sol_object.value(fe_herm.quadrature_point(q));
+ err_sq += sol_at_point * sol_at_point * fe_herm.JxW(q);
+ }
+ }
+
+ err_sq = std::sqrt(err_sq);
+ deallog.depth_file(2);
+
+ deallog << "Test polynomial:" << std::endl;
+ deallog << sol_object.get_function_string() << std::endl;
+ deallog << std::endl;
+
+ deallog << "Interpolation error:" << std::endl;
+ deallog << err_sq << "\n\n" << std::endl;
+ deallog.pop();
+}
+
+
+
+int
+main()
+{
+ std::ofstream logfile("output");
+ deallog << std::setprecision(8) << std::fixed;
+ deallog.attach(logfile);
+
+ test_fe_on_domain<1>(0);
+ test_fe_on_domain<1>(1);
+ test_fe_on_domain<1>(2);
+ test_fe_on_domain<1>(3);
+
+ test_fe_on_domain<2>(0);
+ test_fe_on_domain<2>(1);
+ test_fe_on_domain<2>(2);
+ test_fe_on_domain<2>(3);
+
+ test_fe_on_domain<3>(0);
+ test_fe_on_domain<3>(1);
+
+ return 0;
+}
--- /dev/null
+
+DEAL::
+DEAL:Cell-1d-Hermite-0::Test polynomial:
+DEAL:Cell-1d-Hermite-0::X + 0.5 X^2 - X^3
+DEAL:Cell-1d-Hermite-0::
+DEAL:Cell-1d-Hermite-0::Interpolation error:
+DEAL:Cell-1d-Hermite-0::0.10346989
+
+
+DEAL::
+DEAL:Cell-1d-Hermite-1::Test polynomial:
+DEAL:Cell-1d-Hermite-1::X + 0.5 X^2 - X^3
+DEAL:Cell-1d-Hermite-1::
+DEAL:Cell-1d-Hermite-1::Interpolation error:
+DEAL:Cell-1d-Hermite-1::0.00000000
+
+
+DEAL::
+DEAL:Cell-1d-Hermite-2::Test polynomial:
+DEAL:Cell-1d-Hermite-2::X + 0.5 X^2 - X^3
+DEAL:Cell-1d-Hermite-2::
+DEAL:Cell-1d-Hermite-2::Interpolation error:
+DEAL:Cell-1d-Hermite-2::0.00000000
+
+
+DEAL::
+DEAL:Cell-1d-Hermite-3::Test polynomial:
+DEAL:Cell-1d-Hermite-3::X + 0.5 X^2 - X^3
+DEAL:Cell-1d-Hermite-3::
+DEAL:Cell-1d-Hermite-3::Interpolation error:
+DEAL:Cell-1d-Hermite-3::0.00000000
+
+
+DEAL::
+DEAL:Cell-2d-Hermite-0::Test polynomial:
+DEAL:Cell-2d-Hermite-0::(X + 0.5 X^2 - X^3)(1 - Y^2)
+DEAL:Cell-2d-Hermite-0::
+DEAL:Cell-2d-Hermite-0::Interpolation error:
+DEAL:Cell-2d-Hermite-0::0.09659442
+
+
+DEAL::
+DEAL:Cell-2d-Hermite-1::Test polynomial:
+DEAL:Cell-2d-Hermite-1::(X + 0.5 X^2 - X^3)(1 - Y^2)
+DEAL:Cell-2d-Hermite-1::
+DEAL:Cell-2d-Hermite-1::Interpolation error:
+DEAL:Cell-2d-Hermite-1::0.00000000
+
+
+DEAL::
+DEAL:Cell-2d-Hermite-2::Test polynomial:
+DEAL:Cell-2d-Hermite-2::(X + 0.5 X^2 - X^3)(1 - Y^2)
+DEAL:Cell-2d-Hermite-2::
+DEAL:Cell-2d-Hermite-2::Interpolation error:
+DEAL:Cell-2d-Hermite-2::0.00000000
+
+
+DEAL::
+DEAL:Cell-2d-Hermite-3::Test polynomial:
+DEAL:Cell-2d-Hermite-3::(X + 0.5 X^2 - X^3)(1 - Y^2)
+DEAL:Cell-2d-Hermite-3::
+DEAL:Cell-2d-Hermite-3::Interpolation error:
+DEAL:Cell-2d-Hermite-3::0.00000000
+
+
+DEAL::
+DEAL:Cell-3d-Hermite-0::Test polynomial:
+DEAL:Cell-3d-Hermite-0::(X + 0.5 X^2 - X^3)(1 - Y^2)Z
+DEAL:Cell-3d-Hermite-0::
+DEAL:Cell-3d-Hermite-0::Interpolation error:
+DEAL:Cell-3d-Hermite-0::0.05105713
+
+
+DEAL::
+DEAL:Cell-3d-Hermite-1::Test polynomial:
+DEAL:Cell-3d-Hermite-1::(X + 0.5 X^2 - X^3)(1 - Y^2)Z
+DEAL:Cell-3d-Hermite-1::
+DEAL:Cell-3d-Hermite-1::Interpolation error:
+DEAL:Cell-3d-Hermite-1::0.00000000
+
+
--- /dev/null
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2021 - 2021 by the deal.II authors
+//
+// This file is part of the deal.II library.
+//
+// The deal.II library is free software; you can use it, redistribute
+// it, and/or modify it under the terms of the GNU Lesser General
+// Public License as published by the Free Software Foundation; either
+// version 2.1 of the License, or (at your option) any later version.
+// The full text of the license can be found in the file LICENSE.md at
+// the top level directory of deal.II.
+//
+// ---------------------------------------------------------------------
+
+#include <deal.II/base/config.h>
+
+#include <deal.II/base/function.h>
+#include <deal.II/base/quadrature_lib.h>
+#include <deal.II/base/utilities.h>
+
+#include <deal.II/dofs/dof_accessor.h>
+#include <deal.II/dofs/dof_handler.h>
+#include <deal.II/dofs/dof_tools.h>
+
+#include <deal.II/fe/fe_hermite.h>
+#include <deal.II/fe/fe_q.h>
+#include <deal.II/fe/fe_values.h>
+#include <deal.II/fe/mapping_cartesian.h>
+
+#include <deal.II/grid/grid_generator.h>
+#include <deal.II/grid/tria.h>
+#include <deal.II/grid/tria_accessor.h>
+#include <deal.II/grid/tria_iterator.h>
+
+#include <deal.II/lac/affine_constraints.h>
+#include <deal.II/lac/dynamic_sparsity_pattern.h>
+#include <deal.II/lac/sparse_direct.h>
+#include <deal.II/lac/sparse_matrix.h>
+#include <deal.II/lac/vector.h>
+
+#include <deal.II/numerics/data_out.h>
+#include <deal.II/numerics/matrix_tools.h>
+#include <deal.II/numerics/vector_tools.h>
+
+#include <cmath>
+#include <fstream>
+#include <iterator>
+#include <sstream>
+#include <vector>
+
+
+
+/*
+ * Test case for Hermite with a time-stepping method for $u_{tt}=u_{xx}$
+ * to check that the stability conditions have transferred correctly. Uses a
+ * manufactured solution of a plane wave entering the domain at $x=0$
+ * and leaving at $x=3$. The solution uses $y = x-t-0.5$ as a local
+ * coordinate and has the following Gaussian bell shape to avoid
+ * discontinuities in derivatives:
+ *
+ * @f{align*}{
+ * u(x,t) = exp(-a y^2),
+ * @f}
+ *
+ * where $a$ is some predefined constant. The right-most tail of the wave
+ * is already in the domain at $t=0$. The solution is chosen so that initial
+ * conditions are not uniformly zero and the wave is currently entering
+ * the domain through the boundary. This allows both the use of
+ * boundary projection and initial projection for both value and
+ * derivatives to be tested.
+ */
+using namespace dealii;
+
+template <int dim>
+class Solution : public Function<dim>
+{
+public:
+ Solution(const Solution &sol) = default;
+ Solution(double initial_time = 0, double alpha = 40, bool boundary = false)
+ : alpha(alpha)
+ , current_time(initial_time)
+ , zero_boundary(boundary){};
+
+
+
+ double inline update_time(double time_step)
+ {
+ return this->current_time += time_step;
+ }
+
+
+
+ double inline get_time() const
+ {
+ return this->current_time;
+ }
+
+
+
+ bool inline is_boundary_zero() const
+ {
+ return this->zero_boundary;
+ }
+
+
+
+ virtual double
+ value(const Point<dim> &p, const unsigned int component = 0) const override
+ {
+ (void)component;
+ if (zero_boundary)
+ {
+ double return_val = 1.;
+ for (unsigned int i = 0; i < dim; ++i)
+ return_val *= std::sin(numbers::PI * p(i));
+ return_val *= std::cos(numbers::PI * std::sqrt(dim) * current_time);
+ return return_val;
+ }
+ else
+ {
+ double y = p(0) + 0.5 - current_time;
+ return std::exp(-alpha * y * y);
+ }
+ }
+
+
+
+ double alpha;
+
+private:
+ double current_time;
+ bool zero_boundary;
+};
+
+
+
+template <int dim>
+class SolutionDerivative : public Function<dim>
+{
+public:
+ SolutionDerivative() = delete;
+ SolutionDerivative(const Solution<dim> &parent)
+ : parent_solution(parent){};
+
+
+
+ virtual double
+ value(const Point<dim> &p, const unsigned int component = 0) const override
+ {
+ (void)component;
+ if (parent_solution.is_boundary_zero())
+ {
+ double return_val = -numbers::PI * std::sqrt(dim);
+ for (unsigned int i = 0; i < dim; ++i)
+ return_val *= std::sin(numbers::PI * p(i));
+ return_val *= std::sin(numbers::PI * std::sqrt(dim) *
+ this->parent_solution.get_time());
+ return return_val;
+ }
+ else
+ {
+ double y = p(0) + 0.5 - this->parent_solution.get_time();
+ return 2 * (parent_solution.alpha) * y *
+ std::exp(-(parent_solution.alpha) * y * y);
+ }
+ }
+
+
+
+private:
+ Solution<dim> parent_solution;
+};
+
+
+
+template <int dim>
+double
+get_cfl_number(const unsigned int regularity)
+{
+ double c_eff = 1.0;
+ return c_eff / (std::sqrt(12.0 * dim) * (regularity + 1));
+}
+
+
+
+template <int dim>
+double
+l2_error(const Mapping<dim> & mapping,
+ const DoFHandler<dim> &dof,
+ const Quadrature<dim> &quadrature,
+ const Function<dim> & solution,
+ const Vector<double> & fe_solution)
+{
+ FEValues<dim> herm_vals(mapping,
+ dof.get_fe(),
+ quadrature,
+ update_values | update_quadrature_points |
+ update_JxW_values);
+ std::vector<types::global_dof_index> local_to_global(
+ dof.get_fe().n_dofs_per_cell());
+ double err_sq = 0;
+
+ for (const auto &cell : dof.active_cell_iterators())
+ {
+ herm_vals.reinit(cell);
+ cell->get_dof_indices(local_to_global);
+
+ for (const unsigned int q : herm_vals.quadrature_point_indices())
+ {
+ double diff = solution.value(herm_vals.quadrature_point(q));
+
+ for (const unsigned int i : herm_vals.dof_indices())
+ diff -=
+ fe_solution(local_to_global[i]) * herm_vals.shape_value(i, q);
+
+ err_sq += herm_vals.JxW(q) * diff * diff;
+ }
+ }
+
+ return std::sqrt(err_sq);
+}
+
+
+
+template <int dim>
+void
+test_wave_solver(const double initial_time, const unsigned int regularity)
+{
+ Triangulation<dim> tr;
+ DoFHandler<dim> dof(tr);
+
+ double x_left = 0.0, x_right = 3.0;
+ int divisions = 8;
+ double dx = (x_right - x_left) / divisions;
+ double dt = dx * get_cfl_number<dim>(regularity);
+ double final_time = 4.0;
+
+ GridGenerator::subdivided_hyper_cube(tr, divisions, x_left, x_right);
+
+ MappingCartesian<dim> mapping_h;
+ FE_Hermite<dim> fe_h(regularity);
+ dof.distribute_dofs(fe_h);
+
+ AffineConstraints<double> constraints;
+ constraints.close();
+
+ QGauss<dim> quadrature(2 * regularity + 2);
+ QGauss<dim - 1> face_quadrature(2 * regularity + 2);
+
+ Solution<dim> wave(initial_time);
+ SolutionDerivative<dim> wave_tdev(wave);
+
+ Vector<double> initial_conds_value(dof.n_dofs());
+ Vector<double> iniital_conds_tdev(dof.n_dofs());
+
+ Vector<double> sol_prev(dof.n_dofs());
+ Vector<double> sol_curr(dof.n_dofs());
+ Vector<double> sol_next(dof.n_dofs());
+
+ Vector<double> temp(dof.n_dofs());
+
+ SparsityPattern sp;
+ {
+ DynamicSparsityPattern dsp(dof.n_dofs());
+ DoFTools::make_sparsity_pattern(dof, dsp);
+ sp.copy_from(dsp);
+ }
+
+ SparseMatrix<double> mass, mass_solve, tstep_noninvert;
+ mass.reinit(sp), mass_solve.reinit(sp), tstep_noninvert.reinit(sp);
+
+ MatrixCreator::create_mass_matrix(mapping_h, dof, quadrature, mass);
+ MatrixCreator::create_laplace_matrix(mapping_h,
+ dof,
+ quadrature,
+ tstep_noninvert);
+ tstep_noninvert *= -0.5 * dt * dt;
+ tstep_noninvert.add(1.0, mass);
+
+ VectorTools::project_hermite(mapping_h,
+ dof,
+ constraints,
+ quadrature,
+ wave,
+ sol_curr,
+ false,
+ face_quadrature,
+ true);
+ VectorTools::project_hermite(mapping_h,
+ dof,
+ constraints,
+ quadrature,
+ wave_tdev,
+ sol_prev,
+ false,
+ face_quadrature,
+ true);
+
+ std::map<types::global_dof_index, double> boundary_values;
+ std::map<types::boundary_id, const Function<dim> *> boundary_functions;
+
+ boundary_functions.emplace(std::make_pair(0U, &wave));
+ if (dim == 1)
+ boundary_functions.emplace(std::make_pair(1U, &wave));
+
+ std::vector<double> l2_errors;
+ double max_error;
+ l2_errors.emplace_back(
+ l2_error<dim>(mapping_h, dof, quadrature, wave, sol_curr));
+ max_error = l2_errors[0];
+
+ // Make an initial time-step using
+ // u(t+dt) = u(t) + dt u_t(t) +
+ // 0.5 dt^2 M^(-1)A u(t) + O(dt^3, dt^2 dx^(p+1))
+ tstep_noninvert.vmult(temp, sol_curr);
+ mass.vmult(sol_next, sol_prev);
+ sol_next *= dt;
+ temp += sol_next;
+ sol_next = 0;
+
+ wave.update_time(dt);
+ VectorTools::project_hermite_boundary_values(
+ mapping_h,
+ dof,
+ boundary_functions,
+ face_quadrature,
+ VectorTools::HermiteBoundaryType::hermite_dirichlet,
+ boundary_values);
+
+ mass_solve.copy_from(mass);
+ MatrixTools::apply_boundary_values(
+ boundary_values, mass_solve, sol_next, temp, true);
+
+ SparseDirectUMFPACK mass_inv;
+ mass_inv.initialize(mass_solve);
+ mass_inv.vmult(sol_next, temp);
+
+ sol_prev = sol_curr;
+ sol_curr = sol_next;
+ sol_next = 0;
+
+ boundary_values.clear();
+
+ l2_errors.emplace_back(
+ l2_error<dim>(mapping_h, dof, quadrature, wave, sol_curr));
+ max_error = std::max(max_error, *l2_errors.cend());
+
+ while (wave.get_time() < final_time)
+ {
+ mass.vmult(sol_next, sol_prev);
+ tstep_noninvert.vmult(temp, sol_curr);
+ temp *= 2.0;
+ temp -= sol_next;
+ sol_next = 0;
+
+ wave.update_time(dt);
+ VectorTools::project_hermite_boundary_values(
+ mapping_h,
+ dof,
+ boundary_functions,
+ face_quadrature,
+ VectorTools::HermiteBoundaryType::hermite_dirichlet,
+ boundary_values);
+
+ mass_solve.copy_from(mass);
+ MatrixTools::apply_boundary_values(
+ boundary_values, mass_solve, sol_next, temp, true);
+ mass_inv.vmult(sol_next, temp);
+
+ sol_prev = sol_curr;
+ sol_curr = sol_next;
+ sol_next = 0;
+ boundary_values.clear();
+
+ l2_errors.emplace_back(
+ l2_error<dim>(mapping_h, dof, quadrature, wave, sol_curr));
+ max_error = std::max(max_error, *(--l2_errors.cend()));
+ }
+
+ deallog << std::endl;
+ char fname[50];
+ sprintf(fname, "Cell-%dd-Hermite-%d", dim, regularity);
+ deallog.push(fname);
+
+ deallog << "initial_time: " << initial_time << std::endl;
+ deallog << "Final time: " << final_time << std::endl;
+ deallog << std::endl;
+
+ deallog << "Error at final time: " << l2_errors.back() << std::endl;
+ deallog << "Largest error: " << max_error << std::endl;
+ deallog << std::endl;
+
+ double t = initial_time - dt;
+ deallog << "Error time series:";
+ for (const auto &it : l2_errors)
+ deallog << std::endl << (t += dt) << "\t" << it;
+ deallog << "\n\n" << std::endl;
+
+ deallog.pop();
+}
+
+
+
+int
+main()
+{
+ std::ofstream logfile("output");
+ deallog << std::setprecision(8) << std::fixed;
+ deallog.attach(logfile);
+
+ test_wave_solver<1>(-0.3, 1);
+
+ test_wave_solver<1>(0.0, 1);
+ test_wave_solver<1>(0.0, 2);
+ test_wave_solver<1>(0.0, 3);
+
+ test_wave_solver<2>(0.0, 1);
+ test_wave_solver<2>(0.0, 2);
+ test_wave_solver<2>(0.0, 3);
+
+ return 0;
+}
--- /dev/null
+
+DEAL::
+DEAL:Cell-1d-Hermite-1::initial_time: -0.30000000
+DEAL:Cell-1d-Hermite-1::Final time: 4.00000000
+DEAL:Cell-1d-Hermite-1::
+DEAL:Cell-1d-Hermite-1::Error at final time: 0.22551322
+DEAL:Cell-1d-Hermite-1::Largest error: 0.29178762
+DEAL:Cell-1d-Hermite-1::
+DEAL:Cell-1d-Hermite-1::Error time series:
+DEAL:Cell-1d-Hermite-1::-0.30000000 0.00000000
+DEAL:Cell-1d-Hermite-1::-0.24587341 0.00000000
+DEAL:Cell-1d-Hermite-1::-0.19174682 0.00000000
+DEAL:Cell-1d-Hermite-1::-0.13762024 0.00000001
+DEAL:Cell-1d-Hermite-1::-0.08349365 0.00000012
+DEAL:Cell-1d-Hermite-1::-0.02936706 0.00000121
+DEAL:Cell-1d-Hermite-1::0.02475953 0.00000965
+DEAL:Cell-1d-Hermite-1::0.07888611 0.00005932
+DEAL:Cell-1d-Hermite-1::0.13301270 0.00027892
+DEAL:Cell-1d-Hermite-1::0.18713929 0.00098881
+DEAL:Cell-1d-Hermite-1::0.24126588 0.00257412
+DEAL:Cell-1d-Hermite-1::0.29539247 0.00466748
+DEAL:Cell-1d-Hermite-1::0.34951905 0.00531678
+DEAL:Cell-1d-Hermite-1::0.40364564 0.00619593
+DEAL:Cell-1d-Hermite-1::0.45777223 0.01585813
+DEAL:Cell-1d-Hermite-1::0.51189882 0.02517787
+DEAL:Cell-1d-Hermite-1::0.56602540 0.02474389
+DEAL:Cell-1d-Hermite-1::0.62015199 0.02495361
+DEAL:Cell-1d-Hermite-1::0.67427858 0.04340892
+DEAL:Cell-1d-Hermite-1::0.72840517 0.04933113
+DEAL:Cell-1d-Hermite-1::0.78253175 0.03790667
+DEAL:Cell-1d-Hermite-1::0.83665834 0.05684815
+DEAL:Cell-1d-Hermite-1::0.89078493 0.06880116
+DEAL:Cell-1d-Hermite-1::0.94491152 0.05754642
+DEAL:Cell-1d-Hermite-1::0.99903811 0.07066735
+DEAL:Cell-1d-Hermite-1::1.05316469 0.08660831
+DEAL:Cell-1d-Hermite-1::1.10729128 0.08123024
+DEAL:Cell-1d-Hermite-1::1.16141787 0.08344345
+DEAL:Cell-1d-Hermite-1::1.21554446 0.10236802
+DEAL:Cell-1d-Hermite-1::1.26967104 0.10236021
+DEAL:Cell-1d-Hermite-1::1.32379763 0.09745534
+DEAL:Cell-1d-Hermite-1::1.37792422 0.11594415
+DEAL:Cell-1d-Hermite-1::1.43205081 0.12207087
+DEAL:Cell-1d-Hermite-1::1.48617740 0.11541488
+DEAL:Cell-1d-Hermite-1::1.54030398 0.12715852
+DEAL:Cell-1d-Hermite-1::1.59443057 0.14039456
+DEAL:Cell-1d-Hermite-1::1.64855716 0.13325926
+DEAL:Cell-1d-Hermite-1::1.70268375 0.13647514
+DEAL:Cell-1d-Hermite-1::1.75681033 0.15464753
+DEAL:Cell-1d-Hermite-1::1.81093692 0.15148152
+DEAL:Cell-1d-Hermite-1::1.86506351 0.14814546
+DEAL:Cell-1d-Hermite-1::1.91919010 0.16561086
+DEAL:Cell-1d-Hermite-1::1.97331668 0.16994194
+DEAL:Cell-1d-Hermite-1::2.02744327 0.15871818
+DEAL:Cell-1d-Hermite-1::2.08156986 0.17170082
+DEAL:Cell-1d-Hermite-1::2.13569645 0.18617114
+DEAL:Cell-1d-Hermite-1::2.18982304 0.17166996
+DEAL:Cell-1d-Hermite-1::2.24394962 0.17405838
+DEAL:Cell-1d-Hermite-1::2.29807621 0.19949422
+DEAL:Cell-1d-Hermite-1::2.35220280 0.19352787
+DEAL:Cell-1d-Hermite-1::2.40632939 0.17046018
+DEAL:Cell-1d-Hermite-1::2.46045597 0.19504806
+DEAL:Cell-1d-Hermite-1::2.51458256 0.21953874
+DEAL:Cell-1d-Hermite-1::2.56870915 0.19036599
+DEAL:Cell-1d-Hermite-1::2.62283574 0.16924399
+DEAL:Cell-1d-Hermite-1::2.67696233 0.21545639
+DEAL:Cell-1d-Hermite-1::2.73108891 0.23869446
+DEAL:Cell-1d-Hermite-1::2.78521550 0.18898289
+DEAL:Cell-1d-Hermite-1::2.83934209 0.15034395
+DEAL:Cell-1d-Hermite-1::2.89346868 0.22207063
+DEAL:Cell-1d-Hermite-1::2.94759526 0.26568692
+DEAL:Cell-1d-Hermite-1::3.00172185 0.20477011
+DEAL:Cell-1d-Hermite-1::3.05584844 0.10586578
+DEAL:Cell-1d-Hermite-1::3.10997503 0.20439915
+DEAL:Cell-1d-Hermite-1::3.16410162 0.29178762
+DEAL:Cell-1d-Hermite-1::3.21822820 0.25130384
+DEAL:Cell-1d-Hermite-1::3.27235479 0.12240627
+DEAL:Cell-1d-Hermite-1::3.32648138 0.16318461
+DEAL:Cell-1d-Hermite-1::3.38060797 0.27027831
+DEAL:Cell-1d-Hermite-1::3.43473455 0.27321698
+DEAL:Cell-1d-Hermite-1::3.48886114 0.21608360
+DEAL:Cell-1d-Hermite-1::3.54298773 0.20674395
+DEAL:Cell-1d-Hermite-1::3.59711432 0.22832831
+DEAL:Cell-1d-Hermite-1::3.65124090 0.23187022
+DEAL:Cell-1d-Hermite-1::3.70536749 0.23726211
+DEAL:Cell-1d-Hermite-1::3.75949408 0.23752876
+DEAL:Cell-1d-Hermite-1::3.81362067 0.22555928
+DEAL:Cell-1d-Hermite-1::3.86774726 0.23129945
+DEAL:Cell-1d-Hermite-1::3.92187384 0.23967421
+DEAL:Cell-1d-Hermite-1::3.97600043 0.22630446
+DEAL:Cell-1d-Hermite-1::4.03012702 0.22551322
+
+
+DEAL::
+DEAL:Cell-1d-Hermite-1::initial_time: 0.00000000
+DEAL:Cell-1d-Hermite-1::Final time: 4.00000000
+DEAL:Cell-1d-Hermite-1::
+DEAL:Cell-1d-Hermite-1::Error at final time: 0.22194440
+DEAL:Cell-1d-Hermite-1::Largest error: 0.28630406
+DEAL:Cell-1d-Hermite-1::
+DEAL:Cell-1d-Hermite-1::Error time series:
+DEAL:Cell-1d-Hermite-1::0.00000000 0.00000379
+DEAL:Cell-1d-Hermite-1::0.05412659 0.00002661
+DEAL:Cell-1d-Hermite-1::0.10825318 0.00014214
+DEAL:Cell-1d-Hermite-1::0.16237976 0.00057489
+DEAL:Cell-1d-Hermite-1::0.21650635 0.00173112
+DEAL:Cell-1d-Hermite-1::0.27063294 0.00373809
+DEAL:Cell-1d-Hermite-1::0.32475953 0.00534625
+DEAL:Cell-1d-Hermite-1::0.37888611 0.00498009
+DEAL:Cell-1d-Hermite-1::0.43301270 0.01070557
+DEAL:Cell-1d-Hermite-1::0.48713929 0.02173495
+DEAL:Cell-1d-Hermite-1::0.54126588 0.02629731
+DEAL:Cell-1d-Hermite-1::0.59539247 0.02255269
+DEAL:Cell-1d-Hermite-1::0.64951905 0.03433388
+DEAL:Cell-1d-Hermite-1::0.70364564 0.04993792
+DEAL:Cell-1d-Hermite-1::0.75777223 0.04243038
+DEAL:Cell-1d-Hermite-1::0.81189882 0.04471166
+DEAL:Cell-1d-Hermite-1::0.86602540 0.06744789
+DEAL:Cell-1d-Hermite-1::0.92015199 0.06274341
+DEAL:Cell-1d-Hermite-1::0.97427858 0.06080041
+DEAL:Cell-1d-Hermite-1::1.02840517 0.08201198
+DEAL:Cell-1d-Hermite-1::1.08253175 0.08532550
+DEAL:Cell-1d-Hermite-1::1.13665834 0.07904496
+DEAL:Cell-1d-Hermite-1::1.19078493 0.09417106
+DEAL:Cell-1d-Hermite-1::1.24491152 0.10557570
+DEAL:Cell-1d-Hermite-1::1.29903811 0.09691563
+DEAL:Cell-1d-Hermite-1::1.35316469 0.10642708
+DEAL:Cell-1d-Hermite-1::1.40729128 0.12245252
+DEAL:Cell-1d-Hermite-1::1.46141787 0.11763704
+DEAL:Cell-1d-Hermite-1::1.51554446 0.11924583
+DEAL:Cell-1d-Hermite-1::1.56967104 0.13662795
+DEAL:Cell-1d-Hermite-1::1.62379763 0.13811986
+DEAL:Cell-1d-Hermite-1::1.67792422 0.13122584
+DEAL:Cell-1d-Hermite-1::1.73205081 0.14752365
+DEAL:Cell-1d-Hermite-1::1.78617740 0.15571459
+DEAL:Cell-1d-Hermite-1::1.84030398 0.14673015
+DEAL:Cell-1d-Hermite-1::1.89443057 0.15683327
+DEAL:Cell-1d-Hermite-1::1.94855716 0.17136182
+DEAL:Cell-1d-Hermite-1::2.00268375 0.16313100
+DEAL:Cell-1d-Hermite-1::2.05681033 0.16206217
+DEAL:Cell-1d-Hermite-1::2.11093692 0.18307418
+DEAL:Cell-1d-Hermite-1::2.16506351 0.18022562
+DEAL:Cell-1d-Hermite-1::2.21919010 0.16754828
+DEAL:Cell-1d-Hermite-1::2.27331668 0.18901785
+DEAL:Cell-1d-Hermite-1::2.32744327 0.20162537
+DEAL:Cell-1d-Hermite-1::2.38156986 0.17819658
+DEAL:Cell-1d-Hermite-1::2.43569645 0.17786799
+DEAL:Cell-1d-Hermite-1::2.48982304 0.21415097
+DEAL:Cell-1d-Hermite-1::2.54394962 0.20915575
+DEAL:Cell-1d-Hermite-1::2.59807621 0.17033018
+DEAL:Cell-1d-Hermite-1::2.65220280 0.19040196
+DEAL:Cell-1d-Hermite-1::2.70632939 0.23663756
+DEAL:Cell-1d-Hermite-1::2.76045597 0.21925156
+DEAL:Cell-1d-Hermite-1::2.81458256 0.15464459
+DEAL:Cell-1d-Hermite-1::2.86870915 0.18311201
+DEAL:Cell-1d-Hermite-1::2.92283574 0.25692924
+DEAL:Cell-1d-Hermite-1::2.97696233 0.24576478
+DEAL:Cell-1d-Hermite-1::3.03108891 0.14044054
+DEAL:Cell-1d-Hermite-1::3.08521550 0.14294399
+DEAL:Cell-1d-Hermite-1::3.13934209 0.26537302
+DEAL:Cell-1d-Hermite-1::3.19346868 0.28630406
+DEAL:Cell-1d-Hermite-1::3.24759526 0.18300694
+DEAL:Cell-1d-Hermite-1::3.30172185 0.11037732
+DEAL:Cell-1d-Hermite-1::3.35584844 0.23169162
+DEAL:Cell-1d-Hermite-1::3.40997503 0.28493614
+DEAL:Cell-1d-Hermite-1::3.46410162 0.24224631
+DEAL:Cell-1d-Hermite-1::3.51822820 0.20200481
+DEAL:Cell-1d-Hermite-1::3.57235479 0.22004174
+DEAL:Cell-1d-Hermite-1::3.62648138 0.23161944
+DEAL:Cell-1d-Hermite-1::3.68060797 0.23382735
+DEAL:Cell-1d-Hermite-1::3.73473455 0.23969710
+DEAL:Cell-1d-Hermite-1::3.78886114 0.23069418
+DEAL:Cell-1d-Hermite-1::3.84298773 0.22575034
+DEAL:Cell-1d-Hermite-1::3.89711432 0.23840411
+DEAL:Cell-1d-Hermite-1::3.95124090 0.23384186
+DEAL:Cell-1d-Hermite-1::4.00536749 0.22194440
+
+
+DEAL::
+DEAL:Cell-1d-Hermite-2::initial_time: 0.00000000
+DEAL:Cell-1d-Hermite-2::Final time: 4.00000000
+DEAL:Cell-1d-Hermite-2::
+DEAL:Cell-1d-Hermite-2::Error at final time: 0.07706055
+DEAL:Cell-1d-Hermite-2::Largest error: 0.10410102
+DEAL:Cell-1d-Hermite-2::
+DEAL:Cell-1d-Hermite-2::Error time series:
+DEAL:Cell-1d-Hermite-2::0.00000000 0.00000079
+DEAL:Cell-1d-Hermite-2::0.03608439 0.00000350
+DEAL:Cell-1d-Hermite-2::0.07216878 0.00001102
+DEAL:Cell-1d-Hermite-2::0.10825318 0.00002908
+DEAL:Cell-1d-Hermite-2::0.14433757 0.00006773
+DEAL:Cell-1d-Hermite-2::0.18042196 0.00013664
+DEAL:Cell-1d-Hermite-2::0.21650635 0.00023134
+DEAL:Cell-1d-Hermite-2::0.25259074 0.00031909
+DEAL:Cell-1d-Hermite-2::0.28867513 0.00038769
+DEAL:Cell-1d-Hermite-2::0.32475953 0.00065919
+DEAL:Cell-1d-Hermite-2::0.36084392 0.00132611
+DEAL:Cell-1d-Hermite-2::0.39692831 0.00208499
+DEAL:Cell-1d-Hermite-2::0.43301270 0.00236175
+DEAL:Cell-1d-Hermite-2::0.46909709 0.00211023
+DEAL:Cell-1d-Hermite-2::0.50518149 0.00342575
+DEAL:Cell-1d-Hermite-2::0.54126588 0.00593364
+DEAL:Cell-1d-Hermite-2::0.57735027 0.00701837
+DEAL:Cell-1d-Hermite-2::0.61343466 0.00602703
+DEAL:Cell-1d-Hermite-2::0.64951905 0.00671738
+DEAL:Cell-1d-Hermite-2::0.68560344 0.00968483
+DEAL:Cell-1d-Hermite-2::0.72168784 0.01055455
+DEAL:Cell-1d-Hermite-2::0.75777223 0.01010022
+DEAL:Cell-1d-Hermite-2::0.79385662 0.01120774
+DEAL:Cell-1d-Hermite-2::0.82994101 0.01176692
+DEAL:Cell-1d-Hermite-2::0.86602540 0.01233282
+DEAL:Cell-1d-Hermite-2::0.90210980 0.01514458
+DEAL:Cell-1d-Hermite-2::0.93819419 0.01620213
+DEAL:Cell-1d-Hermite-2::0.97427858 0.01531505
+DEAL:Cell-1d-Hermite-2::1.01036297 0.01724324
+DEAL:Cell-1d-Hermite-2::1.04644736 0.01961836
+DEAL:Cell-1d-Hermite-2::1.08253175 0.01986434
+DEAL:Cell-1d-Hermite-2::1.11861615 0.02065284
+DEAL:Cell-1d-Hermite-2::1.15470054 0.02174152
+DEAL:Cell-1d-Hermite-2::1.19078493 0.02162069
+DEAL:Cell-1d-Hermite-2::1.22686932 0.02254425
+DEAL:Cell-1d-Hermite-2::1.26295371 0.02478217
+DEAL:Cell-1d-Hermite-2::1.29903811 0.02583899
+DEAL:Cell-1d-Hermite-2::1.33512250 0.02597972
+DEAL:Cell-1d-Hermite-2::1.37120689 0.02706890
+DEAL:Cell-1d-Hermite-2::1.40729128 0.02869694
+DEAL:Cell-1d-Hermite-2::1.44337567 0.02983762
+DEAL:Cell-1d-Hermite-2::1.47946006 0.03103853
+DEAL:Cell-1d-Hermite-2::1.51554446 0.03208393
+DEAL:Cell-1d-Hermite-2::1.55162885 0.03221564
+DEAL:Cell-1d-Hermite-2::1.58771324 0.03283204
+DEAL:Cell-1d-Hermite-2::1.62379763 0.03478815
+DEAL:Cell-1d-Hermite-2::1.65988202 0.03627798
+DEAL:Cell-1d-Hermite-2::1.69596642 0.03669089
+DEAL:Cell-1d-Hermite-2::1.73205081 0.03735203
+DEAL:Cell-1d-Hermite-2::1.76813520 0.03863011
+DEAL:Cell-1d-Hermite-2::1.80421959 0.03990360
+DEAL:Cell-1d-Hermite-2::1.84030398 0.04119143
+DEAL:Cell-1d-Hermite-2::1.87638837 0.04235139
+DEAL:Cell-1d-Hermite-2::1.91247277 0.04265379
+DEAL:Cell-1d-Hermite-2::1.94855716 0.04286020
+DEAL:Cell-1d-Hermite-2::1.98464155 0.04442606
+DEAL:Cell-1d-Hermite-2::2.02072594 0.04621990
+DEAL:Cell-1d-Hermite-2::2.05681033 0.04682928
+DEAL:Cell-1d-Hermite-2::2.09289473 0.04722343
+DEAL:Cell-1d-Hermite-2::2.12897912 0.04821039
+DEAL:Cell-1d-Hermite-2::2.16506351 0.04941221
+DEAL:Cell-1d-Hermite-2::2.20114790 0.05074686
+DEAL:Cell-1d-Hermite-2::2.23723229 0.05192298
+DEAL:Cell-1d-Hermite-2::2.27331668 0.05242094
+DEAL:Cell-1d-Hermite-2::2.30940108 0.05262012
+DEAL:Cell-1d-Hermite-2::2.34548547 0.05350399
+DEAL:Cell-1d-Hermite-2::2.38156986 0.05520150
+DEAL:Cell-1d-Hermite-2::2.41765425 0.05658466
+DEAL:Cell-1d-Hermite-2::2.45373864 0.05686476
+DEAL:Cell-1d-Hermite-2::2.48982304 0.05690150
+DEAL:Cell-1d-Hermite-2::2.52590743 0.05822077
+DEAL:Cell-1d-Hermite-2::2.56199182 0.06034438
+DEAL:Cell-1d-Hermite-2::2.59807621 0.06101610
+DEAL:Cell-1d-Hermite-2::2.63416060 0.06063833
+DEAL:Cell-1d-Hermite-2::2.67024500 0.06195349
+DEAL:Cell-1d-Hermite-2::2.70632939 0.06363212
+DEAL:Cell-1d-Hermite-2::2.74241378 0.06309761
+DEAL:Cell-1d-Hermite-2::2.77849817 0.06359940
+DEAL:Cell-1d-Hermite-2::2.81458256 0.06717150
+DEAL:Cell-1d-Hermite-2::2.85066695 0.06790225
+DEAL:Cell-1d-Hermite-2::2.88675135 0.06430523
+DEAL:Cell-1d-Hermite-2::2.92283574 0.06584277
+DEAL:Cell-1d-Hermite-2::2.95892013 0.07314012
+DEAL:Cell-1d-Hermite-2::2.99500452 0.07293442
+DEAL:Cell-1d-Hermite-2::3.03108891 0.06466273
+DEAL:Cell-1d-Hermite-2::3.06717331 0.06727141
+DEAL:Cell-1d-Hermite-2::3.10325770 0.07985109
+DEAL:Cell-1d-Hermite-2::3.13934209 0.07870921
+DEAL:Cell-1d-Hermite-2::3.17542648 0.06068082
+DEAL:Cell-1d-Hermite-2::3.21151087 0.06031980
+DEAL:Cell-1d-Hermite-2::3.24759526 0.08722120
+DEAL:Cell-1d-Hermite-2::3.28367966 0.09875673
+DEAL:Cell-1d-Hermite-2::3.31976405 0.07646185
+DEAL:Cell-1d-Hermite-2::3.35584844 0.02993443
+DEAL:Cell-1d-Hermite-2::3.39193283 0.04465044
+DEAL:Cell-1d-Hermite-2::3.42801722 0.08799844
+DEAL:Cell-1d-Hermite-2::3.46410162 0.10410102
+DEAL:Cell-1d-Hermite-2::3.50018601 0.09217149
+DEAL:Cell-1d-Hermite-2::3.53627040 0.06899119
+DEAL:Cell-1d-Hermite-2::3.57235479 0.06046581
+DEAL:Cell-1d-Hermite-2::3.60843918 0.07006805
+DEAL:Cell-1d-Hermite-2::3.64452357 0.07947166
+DEAL:Cell-1d-Hermite-2::3.68060797 0.08272664
+DEAL:Cell-1d-Hermite-2::3.71669236 0.08194750
+DEAL:Cell-1d-Hermite-2::3.75277675 0.07949280
+DEAL:Cell-1d-Hermite-2::3.78886114 0.07713430
+DEAL:Cell-1d-Hermite-2::3.82494553 0.07674701
+DEAL:Cell-1d-Hermite-2::3.86102993 0.07832030
+DEAL:Cell-1d-Hermite-2::3.89711432 0.07969775
+DEAL:Cell-1d-Hermite-2::3.93319871 0.07935464
+DEAL:Cell-1d-Hermite-2::3.96928310 0.07781652
+DEAL:Cell-1d-Hermite-2::4.00536749 0.07706055
+
+
+DEAL::
+DEAL:Cell-1d-Hermite-3::initial_time: 0.00000000
+DEAL:Cell-1d-Hermite-3::Final time: 4.00000000
+DEAL:Cell-1d-Hermite-3::
+DEAL:Cell-1d-Hermite-3::Error at final time: 0.04082433
+DEAL:Cell-1d-Hermite-3::Largest error: 0.05554646
+DEAL:Cell-1d-Hermite-3::
+DEAL:Cell-1d-Hermite-3::Error time series:
+DEAL:Cell-1d-Hermite-3::0.00000000 0.00000013
+DEAL:Cell-1d-Hermite-3::0.02706329 0.00000106
+DEAL:Cell-1d-Hermite-3::0.05412659 0.00000192
+DEAL:Cell-1d-Hermite-3::0.08118988 0.00000339
+DEAL:Cell-1d-Hermite-3::0.10825318 0.00000680
+DEAL:Cell-1d-Hermite-3::0.13531647 0.00001315
+DEAL:Cell-1d-Hermite-3::0.16237976 0.00002355
+DEAL:Cell-1d-Hermite-3::0.18944306 0.00004059
+DEAL:Cell-1d-Hermite-3::0.21650635 0.00006774
+DEAL:Cell-1d-Hermite-3::0.24356964 0.00010774
+DEAL:Cell-1d-Hermite-3::0.27063294 0.00016126
+DEAL:Cell-1d-Hermite-3::0.29769623 0.00022431
+DEAL:Cell-1d-Hermite-3::0.32475953 0.00028781
+DEAL:Cell-1d-Hermite-3::0.35182282 0.00034666
+DEAL:Cell-1d-Hermite-3::0.37888611 0.00041709
+DEAL:Cell-1d-Hermite-3::0.40594941 0.00053367
+DEAL:Cell-1d-Hermite-3::0.43301270 0.00070266
+DEAL:Cell-1d-Hermite-3::0.46007600 0.00088517
+DEAL:Cell-1d-Hermite-3::0.48713929 0.00105275
+DEAL:Cell-1d-Hermite-3::0.51420258 0.00124372
+DEAL:Cell-1d-Hermite-3::0.54126588 0.00151985
+DEAL:Cell-1d-Hermite-3::0.56832917 0.00185025
+DEAL:Cell-1d-Hermite-3::0.59539247 0.00213818
+DEAL:Cell-1d-Hermite-3::0.62245576 0.00236018
+DEAL:Cell-1d-Hermite-3::0.64951905 0.00260772
+DEAL:Cell-1d-Hermite-3::0.67658235 0.00295283
+DEAL:Cell-1d-Hermite-3::0.70364564 0.00333712
+DEAL:Cell-1d-Hermite-3::0.73070893 0.00367687
+DEAL:Cell-1d-Hermite-3::0.75777223 0.00398082
+DEAL:Cell-1d-Hermite-3::0.78483552 0.00430999
+DEAL:Cell-1d-Hermite-3::0.81189882 0.00467498
+DEAL:Cell-1d-Hermite-3::0.83896211 0.00503166
+DEAL:Cell-1d-Hermite-3::0.86602540 0.00536745
+DEAL:Cell-1d-Hermite-3::0.89308870 0.00573271
+DEAL:Cell-1d-Hermite-3::0.92015199 0.00615422
+DEAL:Cell-1d-Hermite-3::0.94721529 0.00656828
+DEAL:Cell-1d-Hermite-3::0.97427858 0.00691028
+DEAL:Cell-1d-Hermite-3::1.00134187 0.00722400
+DEAL:Cell-1d-Hermite-3::1.02840517 0.00759937
+DEAL:Cell-1d-Hermite-3::1.05546846 0.00803132
+DEAL:Cell-1d-Hermite-3::1.08253175 0.00843583
+DEAL:Cell-1d-Hermite-3::1.10959505 0.00877928
+DEAL:Cell-1d-Hermite-3::1.13665834 0.00911109
+DEAL:Cell-1d-Hermite-3::1.16372164 0.00948136
+DEAL:Cell-1d-Hermite-3::1.19078493 0.00987363
+DEAL:Cell-1d-Hermite-3::1.21784822 0.01023719
+DEAL:Cell-1d-Hermite-3::1.24491152 0.01057194
+DEAL:Cell-1d-Hermite-3::1.27197481 0.01094101
+DEAL:Cell-1d-Hermite-3::1.29903811 0.01137082
+DEAL:Cell-1d-Hermite-3::1.32610140 0.01179246
+DEAL:Cell-1d-Hermite-3::1.35316469 0.01214131
+DEAL:Cell-1d-Hermite-3::1.38022799 0.01245893
+DEAL:Cell-1d-Hermite-3::1.40729128 0.01283059
+DEAL:Cell-1d-Hermite-3::1.43435458 0.01326027
+DEAL:Cell-1d-Hermite-3::1.46141787 0.01367490
+DEAL:Cell-1d-Hermite-3::1.48848116 0.01403254
+DEAL:Cell-1d-Hermite-3::1.51554446 0.01437175
+DEAL:Cell-1d-Hermite-3::1.54260775 0.01474541
+DEAL:Cell-1d-Hermite-3::1.56967104 0.01513781
+DEAL:Cell-1d-Hermite-3::1.59673434 0.01549510
+DEAL:Cell-1d-Hermite-3::1.62379763 0.01582870
+DEAL:Cell-1d-Hermite-3::1.65086093 0.01621145
+DEAL:Cell-1d-Hermite-3::1.67792422 0.01665244
+DEAL:Cell-1d-Hermite-3::1.70498751 0.01706510
+DEAL:Cell-1d-Hermite-3::1.73205081 0.01739702
+DEAL:Cell-1d-Hermite-3::1.75911410 0.01771044
+DEAL:Cell-1d-Hermite-3::1.78617740 0.01809010
+DEAL:Cell-1d-Hermite-3::1.81324069 0.01852473
+DEAL:Cell-1d-Hermite-3::1.84030398 0.01893298
+DEAL:Cell-1d-Hermite-3::1.86736728 0.01928195
+DEAL:Cell-1d-Hermite-3::1.89443057 0.01962505
+DEAL:Cell-1d-Hermite-3::1.92149386 0.02000856
+DEAL:Cell-1d-Hermite-3::1.94855716 0.02039451
+DEAL:Cell-1d-Hermite-3::1.97562045 0.02073397
+DEAL:Cell-1d-Hermite-3::2.00268375 0.02106831
+DEAL:Cell-1d-Hermite-3::2.02974704 0.02147092
+DEAL:Cell-1d-Hermite-3::2.05681033 0.02191678
+DEAL:Cell-1d-Hermite-3::2.08387363 0.02230770
+DEAL:Cell-1d-Hermite-3::2.11093692 0.02261846
+DEAL:Cell-1d-Hermite-3::2.13800022 0.02293424
+DEAL:Cell-1d-Hermite-3::2.16506351 0.02332899
+DEAL:Cell-1d-Hermite-3::2.19212680 0.02376475
+DEAL:Cell-1d-Hermite-3::2.21919010 0.02415464
+DEAL:Cell-1d-Hermite-3::2.24625339 0.02449133
+DEAL:Cell-1d-Hermite-3::2.27331668 0.02484524
+DEAL:Cell-1d-Hermite-3::2.30037998 0.02523723
+DEAL:Cell-1d-Hermite-3::2.32744327 0.02560455
+DEAL:Cell-1d-Hermite-3::2.35450657 0.02592447
+DEAL:Cell-1d-Hermite-3::2.38156986 0.02627118
+DEAL:Cell-1d-Hermite-3::2.40863315 0.02669515
+DEAL:Cell-1d-Hermite-3::2.43569645 0.02713184
+DEAL:Cell-1d-Hermite-3::2.46275974 0.02749276
+DEAL:Cell-1d-Hermite-3::2.48982304 0.02778895
+DEAL:Cell-1d-Hermite-3::2.51688633 0.02811401
+DEAL:Cell-1d-Hermite-3::2.54394962 0.02852113
+DEAL:Cell-1d-Hermite-3::2.57101292 0.02895240
+DEAL:Cell-1d-Hermite-3::2.59807621 0.02932255
+DEAL:Cell-1d-Hermite-3::2.62513951 0.02964908
+DEAL:Cell-1d-Hermite-3::2.65220280 0.03001553
+DEAL:Cell-1d-Hermite-3::2.67926609 0.03041173
+DEAL:Cell-1d-Hermite-3::2.70632939 0.03074922
+DEAL:Cell-1d-Hermite-3::2.73339268 0.03104704
+DEAL:Cell-1d-Hermite-3::2.76045597 0.03142428
+DEAL:Cell-1d-Hermite-3::2.78751927 0.03188279
+DEAL:Cell-1d-Hermite-3::2.81458256 0.03229029
+DEAL:Cell-1d-Hermite-3::2.84164586 0.03259848
+DEAL:Cell-1d-Hermite-3::2.86870915 0.03289280
+DEAL:Cell-1d-Hermite-3::2.89577244 0.03323668
+DEAL:Cell-1d-Hermite-3::2.92283574 0.03362714
+DEAL:Cell-1d-Hermite-3::2.94989903 0.03405241
+DEAL:Cell-1d-Hermite-3::2.97696233 0.03446531
+DEAL:Cell-1d-Hermite-3::3.00402562 0.03480647
+DEAL:Cell-1d-Hermite-3::3.03108891 0.03511649
+DEAL:Cell-1d-Hermite-3::3.05815221 0.03544782
+DEAL:Cell-1d-Hermite-3::3.08521550 0.03571960
+DEAL:Cell-1d-Hermite-3::3.11227879 0.03595485
+DEAL:Cell-1d-Hermite-3::3.13934209 0.03640778
+DEAL:Cell-1d-Hermite-3::3.16640538 0.03691973
+DEAL:Cell-1d-Hermite-3::3.19346868 0.03674259
+DEAL:Cell-1d-Hermite-3::3.22053197 0.03615301
+DEAL:Cell-1d-Hermite-3::3.24759526 0.03758638
+DEAL:Cell-1d-Hermite-3::3.27465856 0.04215168
+DEAL:Cell-1d-Hermite-3::3.30172185 0.04592158
+DEAL:Cell-1d-Hermite-3::3.32878515 0.04335673
+DEAL:Cell-1d-Hermite-3::3.35584844 0.03188325
+DEAL:Cell-1d-Hermite-3::3.38291173 0.01592185
+DEAL:Cell-1d-Hermite-3::3.40997503 0.02171794
+DEAL:Cell-1d-Hermite-3::3.43703832 0.04091958
+DEAL:Cell-1d-Hermite-3::3.46410162 0.05360480
+DEAL:Cell-1d-Hermite-3::3.49116491 0.05554646
+DEAL:Cell-1d-Hermite-3::3.51822820 0.04773111
+DEAL:Cell-1d-Hermite-3::3.54529150 0.03541331
+DEAL:Cell-1d-Hermite-3::3.57235479 0.02783856
+DEAL:Cell-1d-Hermite-3::3.59941808 0.03103238
+DEAL:Cell-1d-Hermite-3::3.62648138 0.03792959
+DEAL:Cell-1d-Hermite-3::3.65354467 0.04244409
+DEAL:Cell-1d-Hermite-3::3.68060797 0.04372230
+DEAL:Cell-1d-Hermite-3::3.70767126 0.04301233
+DEAL:Cell-1d-Hermite-3::3.73473455 0.04185789
+DEAL:Cell-1d-Hermite-3::3.76179785 0.04114523
+DEAL:Cell-1d-Hermite-3::3.78886114 0.04088794
+DEAL:Cell-1d-Hermite-3::3.81592444 0.04073849
+DEAL:Cell-1d-Hermite-3::3.84298773 0.04058719
+DEAL:Cell-1d-Hermite-3::3.87005102 0.04060270
+DEAL:Cell-1d-Hermite-3::3.89711432 0.04083759
+DEAL:Cell-1d-Hermite-3::3.92417761 0.04105703
+DEAL:Cell-1d-Hermite-3::3.95124090 0.04104496
+DEAL:Cell-1d-Hermite-3::3.97830420 0.04088994
+DEAL:Cell-1d-Hermite-3::4.00536749 0.04082433
+
+
+DEAL::
+DEAL:Cell-2d-Hermite-1::initial_time: 0.00000000
+DEAL:Cell-2d-Hermite-1::Final time: 4.00000000
+DEAL:Cell-2d-Hermite-1::
+DEAL:Cell-2d-Hermite-1::Error at final time: 0.32156763
+DEAL:Cell-2d-Hermite-1::Largest error: 0.43001070
+DEAL:Cell-2d-Hermite-1::
+DEAL:Cell-2d-Hermite-1::Error time series:
+DEAL:Cell-2d-Hermite-1::0.00000000 0.00000652
+DEAL:Cell-2d-Hermite-1::0.03827328 0.00002638
+DEAL:Cell-2d-Hermite-1::0.07654655 0.00009376
+DEAL:Cell-2d-Hermite-1::0.11481983 0.00029061
+DEAL:Cell-2d-Hermite-1::0.15309311 0.00078317
+DEAL:Cell-2d-Hermite-1::0.19136639 0.00182390
+DEAL:Cell-2d-Hermite-1::0.22963966 0.00363246
+DEAL:Cell-2d-Hermite-1::0.26791294 0.00607849
+DEAL:Cell-2d-Hermite-1::0.30618622 0.00828533
+DEAL:Cell-2d-Hermite-1::0.34445950 0.00875752
+DEAL:Cell-2d-Hermite-1::0.38273277 0.00803771
+DEAL:Cell-2d-Hermite-1::0.42100605 0.01417808
+DEAL:Cell-2d-Hermite-1::0.45927933 0.02702538
+DEAL:Cell-2d-Hermite-1::0.49755260 0.03891679
+DEAL:Cell-2d-Hermite-1::0.53582588 0.04339331
+DEAL:Cell-2d-Hermite-1::0.57409916 0.03849728
+DEAL:Cell-2d-Hermite-1::0.61237244 0.03645308
+DEAL:Cell-2d-Hermite-1::0.65064571 0.05534326
+DEAL:Cell-2d-Hermite-1::0.68891899 0.07707373
+DEAL:Cell-2d-Hermite-1::0.72719227 0.07965335
+DEAL:Cell-2d-Hermite-1::0.76546554 0.06119476
+DEAL:Cell-2d-Hermite-1::0.80373882 0.06004240
+DEAL:Cell-2d-Hermite-1::0.84201210 0.09126196
+DEAL:Cell-2d-Hermite-1::0.88028538 0.10736608
+DEAL:Cell-2d-Hermite-1::0.91855865 0.09477774
+DEAL:Cell-2d-Hermite-1::0.95683193 0.08155779
+DEAL:Cell-2d-Hermite-1::0.99510521 0.10075521
+DEAL:Cell-2d-Hermite-1::1.03337849 0.12485682
+DEAL:Cell-2d-Hermite-1::1.07165176 0.12920946
+DEAL:Cell-2d-Hermite-1::1.10992504 0.11774027
+DEAL:Cell-2d-Hermite-1::1.14819832 0.11376554
+DEAL:Cell-2d-Hermite-1::1.18647159 0.13214999
+DEAL:Cell-2d-Hermite-1::1.22474487 0.15130723
+DEAL:Cell-2d-Hermite-1::1.26301815 0.14979576
+DEAL:Cell-2d-Hermite-1::1.30129143 0.13671154
+DEAL:Cell-2d-Hermite-1::1.33956470 0.14240140
+DEAL:Cell-2d-Hermite-1::1.37783798 0.16463547
+DEAL:Cell-2d-Hermite-1::1.41611126 0.17539001
+DEAL:Cell-2d-Hermite-1::1.45438453 0.16907045
+DEAL:Cell-2d-Hermite-1::1.49265781 0.16318702
+DEAL:Cell-2d-Hermite-1::1.53093109 0.17343039
+DEAL:Cell-2d-Hermite-1::1.56920437 0.19134555
+DEAL:Cell-2d-Hermite-1::1.60747764 0.19782572
+DEAL:Cell-2d-Hermite-1::1.64575092 0.18888987
+DEAL:Cell-2d-Hermite-1::1.68402420 0.18457206
+DEAL:Cell-2d-Hermite-1::1.72229748 0.20038963
+DEAL:Cell-2d-Hermite-1::1.76057075 0.21739242
+DEAL:Cell-2d-Hermite-1::1.79884403 0.21691961
+DEAL:Cell-2d-Hermite-1::1.83711731 0.20783227
+DEAL:Cell-2d-Hermite-1::1.87539058 0.21095141
+DEAL:Cell-2d-Hermite-1::1.91366386 0.22743675
+DEAL:Cell-2d-Hermite-1::1.95193714 0.23944871
+DEAL:Cell-2d-Hermite-1::1.99021042 0.23542339
+DEAL:Cell-2d-Hermite-1::2.02848369 0.22470583
+DEAL:Cell-2d-Hermite-1::2.06675697 0.23009024
+DEAL:Cell-2d-Hermite-1::2.10503025 0.25077848
+DEAL:Cell-2d-Hermite-1::2.14330352 0.26072519
+DEAL:Cell-2d-Hermite-1::2.18157680 0.25051399
+DEAL:Cell-2d-Hermite-1::2.21985008 0.24005725
+DEAL:Cell-2d-Hermite-1::2.25812336 0.25063484
+DEAL:Cell-2d-Hermite-1::2.29639663 0.27259303
+DEAL:Cell-2d-Hermite-1::2.33466991 0.28176484
+DEAL:Cell-2d-Hermite-1::2.37294319 0.26848050
+DEAL:Cell-2d-Hermite-1::2.41121647 0.24975041
+DEAL:Cell-2d-Hermite-1::2.44948974 0.25679548
+DEAL:Cell-2d-Hermite-1::2.48776302 0.28666307
+DEAL:Cell-2d-Hermite-1::2.52603630 0.30446099
+DEAL:Cell-2d-Hermite-1::2.56430957 0.29227626
+DEAL:Cell-2d-Hermite-1::2.60258285 0.26358323
+DEAL:Cell-2d-Hermite-1::2.64085613 0.25372291
+DEAL:Cell-2d-Hermite-1::2.67912941 0.28295226
+DEAL:Cell-2d-Hermite-1::2.71740268 0.32286569
+DEAL:Cell-2d-Hermite-1::2.75567596 0.33234916
+DEAL:Cell-2d-Hermite-1::2.79394924 0.29594392
+DEAL:Cell-2d-Hermite-1::2.83222252 0.24102644
+DEAL:Cell-2d-Hermite-1::2.87049579 0.23937040
+DEAL:Cell-2d-Hermite-1::2.90876907 0.30739849
+DEAL:Cell-2d-Hermite-1::2.94704235 0.36999819
+DEAL:Cell-2d-Hermite-1::2.98531562 0.36972113
+DEAL:Cell-2d-Hermite-1::3.02358890 0.29159815
+DEAL:Cell-2d-Hermite-1::3.06186218 0.18561167
+DEAL:Cell-2d-Hermite-1::3.10013546 0.21662533
+DEAL:Cell-2d-Hermite-1::3.13840873 0.34524378
+DEAL:Cell-2d-Hermite-1::3.17668201 0.42296404
+DEAL:Cell-2d-Hermite-1::3.21495529 0.40129460
+DEAL:Cell-2d-Hermite-1::3.25322856 0.28510672
+DEAL:Cell-2d-Hermite-1::3.29150184 0.15874839
+DEAL:Cell-2d-Hermite-1::3.32977512 0.23667008
+DEAL:Cell-2d-Hermite-1::3.36804840 0.37410891
+DEAL:Cell-2d-Hermite-1::3.40632167 0.43001070
+DEAL:Cell-2d-Hermite-1::3.44459495 0.38791941
+DEAL:Cell-2d-Hermite-1::3.48286823 0.29915304
+DEAL:Cell-2d-Hermite-1::3.52114151 0.27034196
+DEAL:Cell-2d-Hermite-1::3.55941478 0.32428978
+DEAL:Cell-2d-Hermite-1::3.59768806 0.36494977
+DEAL:Cell-2d-Hermite-1::3.63596134 0.35471000
+DEAL:Cell-2d-Hermite-1::3.67423461 0.32700939
+DEAL:Cell-2d-Hermite-1::3.71250789 0.33087042
+DEAL:Cell-2d-Hermite-1::3.75078117 0.35510403
+DEAL:Cell-2d-Hermite-1::3.78905445 0.35638217
+DEAL:Cell-2d-Hermite-1::3.82732772 0.33111726
+DEAL:Cell-2d-Hermite-1::3.86560100 0.31891327
+DEAL:Cell-2d-Hermite-1::3.90387428 0.34094476
+DEAL:Cell-2d-Hermite-1::3.94214755 0.36160923
+DEAL:Cell-2d-Hermite-1::3.98042083 0.34978308
+DEAL:Cell-2d-Hermite-1::4.01869411 0.32156763
+
+
+DEAL::
+DEAL:Cell-2d-Hermite-2::initial_time: 0.00000000
+DEAL:Cell-2d-Hermite-2::Final time: 4.00000000
+DEAL:Cell-2d-Hermite-2::
+DEAL:Cell-2d-Hermite-2::Error at final time: 0.07980830
+DEAL:Cell-2d-Hermite-2::Largest error: 0.10747766
+DEAL:Cell-2d-Hermite-2::
+DEAL:Cell-2d-Hermite-2::Error time series:
+DEAL:Cell-2d-Hermite-2::0.00000000 0.00000137
+DEAL:Cell-2d-Hermite-2::0.02551552 0.00000357
+DEAL:Cell-2d-Hermite-2::0.05103104 0.00000872
+DEAL:Cell-2d-Hermite-2::0.07654655 0.00001908
+DEAL:Cell-2d-Hermite-2::0.10206207 0.00003831
+DEAL:Cell-2d-Hermite-2::0.12757759 0.00007146
+DEAL:Cell-2d-Hermite-2::0.15309311 0.00012372
+DEAL:Cell-2d-Hermite-2::0.17860863 0.00019731
+DEAL:Cell-2d-Hermite-2::0.20412415 0.00028604
+DEAL:Cell-2d-Hermite-2::0.22963966 0.00036926
+DEAL:Cell-2d-Hermite-2::0.25515518 0.00041213
+DEAL:Cell-2d-Hermite-2::0.28067070 0.00040071
+DEAL:Cell-2d-Hermite-2::0.30618622 0.00050221
+DEAL:Cell-2d-Hermite-2::0.33170174 0.00099204
+DEAL:Cell-2d-Hermite-2::0.35721725 0.00179602
+DEAL:Cell-2d-Hermite-2::0.38273277 0.00269388
+DEAL:Cell-2d-Hermite-2::0.40824829 0.00335999
+DEAL:Cell-2d-Hermite-2::0.43376381 0.00342318
+DEAL:Cell-2d-Hermite-2::0.45927933 0.00278520
+DEAL:Cell-2d-Hermite-2::0.48479484 0.00278833
+DEAL:Cell-2d-Hermite-2::0.51031036 0.00516359
+DEAL:Cell-2d-Hermite-2::0.53582588 0.00822250
+DEAL:Cell-2d-Hermite-2::0.56134140 0.01026242
+DEAL:Cell-2d-Hermite-2::0.58685692 0.01017025
+DEAL:Cell-2d-Hermite-2::0.61237244 0.00802060
+DEAL:Cell-2d-Hermite-2::0.63788795 0.00687779
+DEAL:Cell-2d-Hermite-2::0.66340347 0.01020471
+DEAL:Cell-2d-Hermite-2::0.68891899 0.01408288
+DEAL:Cell-2d-Hermite-2::0.71443451 0.01534242
+DEAL:Cell-2d-Hermite-2::0.73995003 0.01379270
+DEAL:Cell-2d-Hermite-2::0.76546554 0.01213663
+DEAL:Cell-2d-Hermite-2::0.79098106 0.01318664
+DEAL:Cell-2d-Hermite-2::0.81649658 0.01483446
+DEAL:Cell-2d-Hermite-2::0.84201210 0.01479842
+DEAL:Cell-2d-Hermite-2::0.86752762 0.01488513
+DEAL:Cell-2d-Hermite-2::0.89304314 0.01764482
+DEAL:Cell-2d-Hermite-2::0.91855865 0.02064634
+DEAL:Cell-2d-Hermite-2::0.94407417 0.02055867
+DEAL:Cell-2d-Hermite-2::0.96958969 0.01792072
+DEAL:Cell-2d-Hermite-2::0.99510521 0.01777156
+DEAL:Cell-2d-Hermite-2::1.02062073 0.02198754
+DEAL:Cell-2d-Hermite-2::1.04613624 0.02551651
+DEAL:Cell-2d-Hermite-2::1.07165176 0.02553176
+DEAL:Cell-2d-Hermite-2::1.09716728 0.02374909
+DEAL:Cell-2d-Hermite-2::1.12268280 0.02364356
+DEAL:Cell-2d-Hermite-2::1.14819832 0.02510866
+DEAL:Cell-2d-Hermite-2::1.17371384 0.02538642
+DEAL:Cell-2d-Hermite-2::1.19922935 0.02433078
+DEAL:Cell-2d-Hermite-2::1.22474487 0.02462598
+DEAL:Cell-2d-Hermite-2::1.25026039 0.02712716
+DEAL:Cell-2d-Hermite-2::1.27577591 0.02929490
+DEAL:Cell-2d-Hermite-2::1.30129143 0.02938604
+DEAL:Cell-2d-Hermite-2::1.32680694 0.02836011
+DEAL:Cell-2d-Hermite-2::1.35232246 0.02839680
+DEAL:Cell-2d-Hermite-2::1.37783798 0.03011885
+DEAL:Cell-2d-Hermite-2::1.40335350 0.03221022
+DEAL:Cell-2d-Hermite-2::1.42886902 0.03351702
+DEAL:Cell-2d-Hermite-2::1.45438453 0.03402961
+DEAL:Cell-2d-Hermite-2::1.47990005 0.03438241
+DEAL:Cell-2d-Hermite-2::1.50541557 0.03488979
+DEAL:Cell-2d-Hermite-2::1.53093109 0.03516639
+DEAL:Cell-2d-Hermite-2::1.55644661 0.03494372
+DEAL:Cell-2d-Hermite-2::1.58196213 0.03499263
+DEAL:Cell-2d-Hermite-2::1.60747764 0.03637687
+DEAL:Cell-2d-Hermite-2::1.63299316 0.03861022
+DEAL:Cell-2d-Hermite-2::1.65850868 0.03999704
+DEAL:Cell-2d-Hermite-2::1.68402420 0.03985465
+DEAL:Cell-2d-Hermite-2::1.70953972 0.03936377
+DEAL:Cell-2d-Hermite-2::1.73505523 0.04001245
+DEAL:Cell-2d-Hermite-2::1.76057075 0.04164863
+DEAL:Cell-2d-Hermite-2::1.78608627 0.04313856
+DEAL:Cell-2d-Hermite-2::1.81160179 0.04411134
+DEAL:Cell-2d-Hermite-2::1.83711731 0.04495126
+DEAL:Cell-2d-Hermite-2::1.86263283 0.04573893
+DEAL:Cell-2d-Hermite-2::1.88814834 0.04608056
+DEAL:Cell-2d-Hermite-2::1.91366386 0.04588034
+DEAL:Cell-2d-Hermite-2::1.93917938 0.04573375
+DEAL:Cell-2d-Hermite-2::1.96469490 0.04643044
+DEAL:Cell-2d-Hermite-2::1.99021042 0.04808774
+DEAL:Cell-2d-Hermite-2::2.01572593 0.04986456
+DEAL:Cell-2d-Hermite-2::2.04124145 0.05072489
+DEAL:Cell-2d-Hermite-2::2.06675697 0.05054265
+DEAL:Cell-2d-Hermite-2::2.09227249 0.05033646
+DEAL:Cell-2d-Hermite-2::2.11778801 0.05109038
+DEAL:Cell-2d-Hermite-2::2.14330352 0.05254021
+DEAL:Cell-2d-Hermite-2::2.16881904 0.05382066
+DEAL:Cell-2d-Hermite-2::2.19433456 0.05475636
+DEAL:Cell-2d-Hermite-2::2.21985008 0.05566104
+DEAL:Cell-2d-Hermite-2::2.24536560 0.05634608
+DEAL:Cell-2d-Hermite-2::2.27088112 0.05633320
+DEAL:Cell-2d-Hermite-2::2.29639663 0.05594201
+DEAL:Cell-2d-Hermite-2::2.32191215 0.05621079
+DEAL:Cell-2d-Hermite-2::2.34742767 0.05748527
+DEAL:Cell-2d-Hermite-2::2.37294319 0.05896455
+DEAL:Cell-2d-Hermite-2::2.39845871 0.05993348
+DEAL:Cell-2d-Hermite-2::2.42397422 0.06051556
+DEAL:Cell-2d-Hermite-2::2.44948974 0.06100559
+DEAL:Cell-2d-Hermite-2::2.47500526 0.06125957
+DEAL:Cell-2d-Hermite-2::2.50052078 0.06126451
+DEAL:Cell-2d-Hermite-2::2.52603630 0.06176783
+DEAL:Cell-2d-Hermite-2::2.55155182 0.06349496
+DEAL:Cell-2d-Hermite-2::2.57706733 0.06578571
+DEAL:Cell-2d-Hermite-2::2.60258285 0.06690670
+DEAL:Cell-2d-Hermite-2::2.62809837 0.06596603
+DEAL:Cell-2d-Hermite-2::2.65361389 0.06430668
+DEAL:Cell-2d-Hermite-2::2.67912941 0.06456408
+DEAL:Cell-2d-Hermite-2::2.70464492 0.06732615
+DEAL:Cell-2d-Hermite-2::2.73016044 0.06986321
+DEAL:Cell-2d-Hermite-2::2.75567596 0.06954189
+DEAL:Cell-2d-Hermite-2::2.78119148 0.06733624
+DEAL:Cell-2d-Hermite-2::2.80670700 0.06730561
+DEAL:Cell-2d-Hermite-2::2.83222252 0.07104734
+DEAL:Cell-2d-Hermite-2::2.85773803 0.07454290
+DEAL:Cell-2d-Hermite-2::2.88325355 0.07345912
+DEAL:Cell-2d-Hermite-2::2.90876907 0.06887811
+DEAL:Cell-2d-Hermite-2::2.93428459 0.06757282
+DEAL:Cell-2d-Hermite-2::2.95980011 0.07356032
+DEAL:Cell-2d-Hermite-2::2.98531562 0.08083470
+DEAL:Cell-2d-Hermite-2::3.01083114 0.08132676
+DEAL:Cell-2d-Hermite-2::3.03634666 0.07340466
+DEAL:Cell-2d-Hermite-2::3.06186218 0.06501415
+DEAL:Cell-2d-Hermite-2::3.08737770 0.06869908
+DEAL:Cell-2d-Hermite-2::3.11289321 0.08190171
+DEAL:Cell-2d-Hermite-2::3.13840873 0.09012724
+DEAL:Cell-2d-Hermite-2::3.16392425 0.08463291
+DEAL:Cell-2d-Hermite-2::3.18943977 0.06747894
+DEAL:Cell-2d-Hermite-2::3.21495529 0.05620997
+DEAL:Cell-2d-Hermite-2::3.24047081 0.07068417
+DEAL:Cell-2d-Hermite-2::3.26598632 0.09373355
+DEAL:Cell-2d-Hermite-2::3.29150184 0.10442693
+DEAL:Cell-2d-Hermite-2::3.31701736 0.09467905
+DEAL:Cell-2d-Hermite-2::3.34253288 0.06597514
+DEAL:Cell-2d-Hermite-2::3.36804840 0.03287997
+DEAL:Cell-2d-Hermite-2::3.39356391 0.04530085
+DEAL:Cell-2d-Hermite-2::3.41907943 0.08024988
+DEAL:Cell-2d-Hermite-2::3.44459495 0.10307639
+DEAL:Cell-2d-Hermite-2::3.47011047 0.10747766
+DEAL:Cell-2d-Hermite-2::3.49562599 0.09574825
+DEAL:Cell-2d-Hermite-2::3.52114151 0.07663236
+DEAL:Cell-2d-Hermite-2::3.54665702 0.06360989
+DEAL:Cell-2d-Hermite-2::3.57217254 0.06540116
+DEAL:Cell-2d-Hermite-2::3.59768806 0.07469778
+DEAL:Cell-2d-Hermite-2::3.62320358 0.08201083
+DEAL:Cell-2d-Hermite-2::3.64871910 0.08480954
+DEAL:Cell-2d-Hermite-2::3.67423461 0.08465523
+DEAL:Cell-2d-Hermite-2::3.69975013 0.08383623
+DEAL:Cell-2d-Hermite-2::3.72526565 0.08331071
+DEAL:Cell-2d-Hermite-2::3.75078117 0.08260633
+DEAL:Cell-2d-Hermite-2::3.77629669 0.08127576
+DEAL:Cell-2d-Hermite-2::3.80181220 0.07990209
+DEAL:Cell-2d-Hermite-2::3.82732772 0.07948131
+DEAL:Cell-2d-Hermite-2::3.85284324 0.08022277
+DEAL:Cell-2d-Hermite-2::3.87835876 0.08148325
+DEAL:Cell-2d-Hermite-2::3.90387428 0.08255161
+DEAL:Cell-2d-Hermite-2::3.92938980 0.08293386
+DEAL:Cell-2d-Hermite-2::3.95490531 0.08231562
+DEAL:Cell-2d-Hermite-2::3.98042083 0.08093641
+DEAL:Cell-2d-Hermite-2::4.00593635 0.07980830
+
+
+DEAL::
+DEAL:Cell-2d-Hermite-3::initial_time: 0.00000000
+DEAL:Cell-2d-Hermite-3::Final time: 4.00000000
+DEAL:Cell-2d-Hermite-3::
+DEAL:Cell-2d-Hermite-3::Error at final time: 0.03515019
+DEAL:Cell-2d-Hermite-3::Largest error: 0.04821598
+DEAL:Cell-2d-Hermite-3::
+DEAL:Cell-2d-Hermite-3::Error time series:
+DEAL:Cell-2d-Hermite-3::0.00000000 0.00000022
+DEAL:Cell-2d-Hermite-3::0.01913664 0.00000073
+DEAL:Cell-2d-Hermite-3::0.03827328 0.00000147
+DEAL:Cell-2d-Hermite-3::0.05740992 0.00000232
+DEAL:Cell-2d-Hermite-3::0.07654655 0.00000349
+DEAL:Cell-2d-Hermite-3::0.09568319 0.00000536
+DEAL:Cell-2d-Hermite-3::0.11481983 0.00000829
+DEAL:Cell-2d-Hermite-3::0.13395647 0.00001249
+DEAL:Cell-2d-Hermite-3::0.15309311 0.00001814
+DEAL:Cell-2d-Hermite-3::0.17222975 0.00002570
+DEAL:Cell-2d-Hermite-3::0.19136639 0.00003625
+DEAL:Cell-2d-Hermite-3::0.21050302 0.00005156
+DEAL:Cell-2d-Hermite-3::0.22963966 0.00007354
+DEAL:Cell-2d-Hermite-3::0.24877630 0.00010350
+DEAL:Cell-2d-Hermite-3::0.26791294 0.00014124
+DEAL:Cell-2d-Hermite-3::0.28704958 0.00018417
+DEAL:Cell-2d-Hermite-3::0.30618622 0.00022693
+DEAL:Cell-2d-Hermite-3::0.32532286 0.00026289
+DEAL:Cell-2d-Hermite-3::0.34445950 0.00028888
+DEAL:Cell-2d-Hermite-3::0.36359613 0.00031399
+DEAL:Cell-2d-Hermite-3::0.38273277 0.00036503
+DEAL:Cell-2d-Hermite-3::0.40186941 0.00046662
+DEAL:Cell-2d-Hermite-3::0.42100605 0.00061038
+DEAL:Cell-2d-Hermite-3::0.44014269 0.00076048
+DEAL:Cell-2d-Hermite-3::0.45927933 0.00087753
+DEAL:Cell-2d-Hermite-3::0.47841597 0.00094235
+DEAL:Cell-2d-Hermite-3::0.49755260 0.00098336
+DEAL:Cell-2d-Hermite-3::0.51668924 0.00108462
+DEAL:Cell-2d-Hermite-3::0.53582588 0.00131194
+DEAL:Cell-2d-Hermite-3::0.55496252 0.00162328
+DEAL:Cell-2d-Hermite-3::0.57409916 0.00191245
+DEAL:Cell-2d-Hermite-3::0.59323580 0.00209406
+DEAL:Cell-2d-Hermite-3::0.61237244 0.00215407
+DEAL:Cell-2d-Hermite-3::0.63150907 0.00217191
+DEAL:Cell-2d-Hermite-3::0.65064571 0.00228180
+DEAL:Cell-2d-Hermite-3::0.66978235 0.00254680
+DEAL:Cell-2d-Hermite-3::0.68891899 0.00289094
+DEAL:Cell-2d-Hermite-3::0.70805563 0.00319408
+DEAL:Cell-2d-Hermite-3::0.72719227 0.00339485
+DEAL:Cell-2d-Hermite-3::0.74632891 0.00351624
+DEAL:Cell-2d-Hermite-3::0.76546554 0.00363288
+DEAL:Cell-2d-Hermite-3::0.78460218 0.00380597
+DEAL:Cell-2d-Hermite-3::0.80373882 0.00403616
+DEAL:Cell-2d-Hermite-3::0.82287546 0.00427891
+DEAL:Cell-2d-Hermite-3::0.84201210 0.00449801
+DEAL:Cell-2d-Hermite-3::0.86114874 0.00470154
+DEAL:Cell-2d-Hermite-3::0.88028538 0.00493330
+DEAL:Cell-2d-Hermite-3::0.89942201 0.00522538
+DEAL:Cell-2d-Hermite-3::0.91855865 0.00555348
+DEAL:Cell-2d-Hermite-3::0.93769529 0.00584644
+DEAL:Cell-2d-Hermite-3::0.95683193 0.00604676
+DEAL:Cell-2d-Hermite-3::0.97596857 0.00616846
+DEAL:Cell-2d-Hermite-3::0.99510521 0.00630068
+DEAL:Cell-2d-Hermite-3::1.01424185 0.00653759
+DEAL:Cell-2d-Hermite-3::1.03337849 0.00688711
+DEAL:Cell-2d-Hermite-3::1.05251512 0.00726053
+DEAL:Cell-2d-Hermite-3::1.07165176 0.00755400
+DEAL:Cell-2d-Hermite-3::1.09078840 0.00772985
+DEAL:Cell-2d-Hermite-3::1.10992504 0.00783452
+DEAL:Cell-2d-Hermite-3::1.12906168 0.00795453
+DEAL:Cell-2d-Hermite-3::1.14819832 0.00814907
+DEAL:Cell-2d-Hermite-3::1.16733496 0.00841236
+DEAL:Cell-2d-Hermite-3::1.18647159 0.00869132
+DEAL:Cell-2d-Hermite-3::1.20560823 0.00893279
+DEAL:Cell-2d-Hermite-3::1.22474487 0.00912142
+DEAL:Cell-2d-Hermite-3::1.24388151 0.00928901
+DEAL:Cell-2d-Hermite-3::1.26301815 0.00949067
+DEAL:Cell-2d-Hermite-3::1.28215479 0.00975827
+DEAL:Cell-2d-Hermite-3::1.30129143 0.01006731
+DEAL:Cell-2d-Hermite-3::1.32042806 0.01035371
+DEAL:Cell-2d-Hermite-3::1.33956470 0.01056964
+DEAL:Cell-2d-Hermite-3::1.35870134 0.01072790
+DEAL:Cell-2d-Hermite-3::1.37783798 0.01089421
+DEAL:Cell-2d-Hermite-3::1.39697462 0.01113061
+DEAL:Cell-2d-Hermite-3::1.41611126 0.01143993
+DEAL:Cell-2d-Hermite-3::1.43524790 0.01176516
+DEAL:Cell-2d-Hermite-3::1.45438453 0.01203980
+DEAL:Cell-2d-Hermite-3::1.47352117 0.01223813
+DEAL:Cell-2d-Hermite-3::1.49265781 0.01238816
+DEAL:Cell-2d-Hermite-3::1.51179445 0.01254624
+DEAL:Cell-2d-Hermite-3::1.53093109 0.01275508
+DEAL:Cell-2d-Hermite-3::1.55006773 0.01301376
+DEAL:Cell-2d-Hermite-3::1.56920437 0.01328116
+DEAL:Cell-2d-Hermite-3::1.58834101 0.01351114
+DEAL:Cell-2d-Hermite-3::1.60747764 0.01369322
+DEAL:Cell-2d-Hermite-3::1.62661428 0.01386542
+DEAL:Cell-2d-Hermite-3::1.64575092 0.01408361
+DEAL:Cell-2d-Hermite-3::1.66488756 0.01436836
+DEAL:Cell-2d-Hermite-3::1.68402420 0.01468031
+DEAL:Cell-2d-Hermite-3::1.70316084 0.01495251
+DEAL:Cell-2d-Hermite-3::1.72229748 0.01514983
+DEAL:Cell-2d-Hermite-3::1.74143411 0.01530026
+DEAL:Cell-2d-Hermite-3::1.76057075 0.01547160
+DEAL:Cell-2d-Hermite-3::1.77970739 0.01571428
+DEAL:Cell-2d-Hermite-3::1.79884403 0.01602059
+DEAL:Cell-2d-Hermite-3::1.81798067 0.01633420
+DEAL:Cell-2d-Hermite-3::1.83711731 0.01659646
+DEAL:Cell-2d-Hermite-3::1.85625395 0.01678877
+DEAL:Cell-2d-Hermite-3::1.87539058 0.01694281
+DEAL:Cell-2d-Hermite-3::1.89452722 0.01711463
+DEAL:Cell-2d-Hermite-3::1.91366386 0.01734030
+DEAL:Cell-2d-Hermite-3::1.93280050 0.01760632
+DEAL:Cell-2d-Hermite-3::1.95193714 0.01786241
+DEAL:Cell-2d-Hermite-3::1.97107378 0.01806856
+DEAL:Cell-2d-Hermite-3::1.99021042 0.01823513
+DEAL:Cell-2d-Hermite-3::2.00934705 0.01841745
+DEAL:Cell-2d-Hermite-3::2.02848369 0.01866509
+DEAL:Cell-2d-Hermite-3::2.04762033 0.01897174
+DEAL:Cell-2d-Hermite-3::2.06675697 0.01927676
+DEAL:Cell-2d-Hermite-3::2.08589361 0.01951794
+DEAL:Cell-2d-Hermite-3::2.10503025 0.01968480
+DEAL:Cell-2d-Hermite-3::2.12416689 0.01982663
+DEAL:Cell-2d-Hermite-3::2.14330352 0.02001180
+DEAL:Cell-2d-Hermite-3::2.16244016 0.02027375
+DEAL:Cell-2d-Hermite-3::2.18157680 0.02058687
+DEAL:Cell-2d-Hermite-3::2.20071344 0.02088873
+DEAL:Cell-2d-Hermite-3::2.21985008 0.02112763
+DEAL:Cell-2d-Hermite-3::2.23898672 0.02130001
+DEAL:Cell-2d-Hermite-3::2.25812336 0.02145214
+DEAL:Cell-2d-Hermite-3::2.27726000 0.02164257
+DEAL:Cell-2d-Hermite-3::2.29639663 0.02189120
+DEAL:Cell-2d-Hermite-3::2.31553327 0.02216070
+DEAL:Cell-2d-Hermite-3::2.33466991 0.02239245
+DEAL:Cell-2d-Hermite-3::2.35380655 0.02256604
+DEAL:Cell-2d-Hermite-3::2.37294319 0.02272400
+DEAL:Cell-2d-Hermite-3::2.39207983 0.02293378
+DEAL:Cell-2d-Hermite-3::2.41121647 0.02322063
+DEAL:Cell-2d-Hermite-3::2.43035310 0.02353937
+DEAL:Cell-2d-Hermite-3::2.44948974 0.02381487
+DEAL:Cell-2d-Hermite-3::2.46862638 0.02400866
+DEAL:Cell-2d-Hermite-3::2.48776302 0.02414842
+DEAL:Cell-2d-Hermite-3::2.50689966 0.02430030
+DEAL:Cell-2d-Hermite-3::2.52603630 0.02451539
+DEAL:Cell-2d-Hermite-3::2.54517294 0.02479635
+DEAL:Cell-2d-Hermite-3::2.56430957 0.02510286
+DEAL:Cell-2d-Hermite-3::2.58344621 0.02538150
+DEAL:Cell-2d-Hermite-3::2.60258285 0.02559825
+DEAL:Cell-2d-Hermite-3::2.62171949 0.02576048
+DEAL:Cell-2d-Hermite-3::2.64085613 0.02591581
+DEAL:Cell-2d-Hermite-3::2.65999277 0.02611827
+DEAL:Cell-2d-Hermite-3::2.67912941 0.02637844
+DEAL:Cell-2d-Hermite-3::2.69826604 0.02664582
+DEAL:Cell-2d-Hermite-3::2.71740268 0.02685483
+DEAL:Cell-2d-Hermite-3::2.73653932 0.02699817
+DEAL:Cell-2d-Hermite-3::2.75567596 0.02714910
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+DEAL:Cell-2d-Hermite-3::2.79394924 0.02773548
+DEAL:Cell-2d-Hermite-3::2.81308588 0.02807283
+DEAL:Cell-2d-Hermite-3::2.83222252 0.02829876
+DEAL:Cell-2d-Hermite-3::2.85135915 0.02840712
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+DEAL:Cell-2d-Hermite-3::3.00445226 0.03023535
+DEAL:Cell-2d-Hermite-3::3.02358890 0.03043037
+DEAL:Cell-2d-Hermite-3::3.04272554 0.03055769
+DEAL:Cell-2d-Hermite-3::3.06186218 0.03067013
+DEAL:Cell-2d-Hermite-3::3.08099882 0.03084031
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+DEAL:Cell-2d-Hermite-3::3.11927209 0.03127554
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+DEAL:Cell-2d-Hermite-3::3.19581865 0.03233547
+DEAL:Cell-2d-Hermite-3::3.21495529 0.03244597
+DEAL:Cell-2d-Hermite-3::3.23409193 0.03237375
+DEAL:Cell-2d-Hermite-3::3.25322856 0.03264654
+DEAL:Cell-2d-Hermite-3::3.27236520 0.03391572
+DEAL:Cell-2d-Hermite-3::3.29150184 0.03605539
+DEAL:Cell-2d-Hermite-3::3.31063848 0.03779862
+DEAL:Cell-2d-Hermite-3::3.32977512 0.03741909
+DEAL:Cell-2d-Hermite-3::3.34891176 0.03368501
+DEAL:Cell-2d-Hermite-3::3.36804840 0.02664789
+DEAL:Cell-2d-Hermite-3::3.38718503 0.01896223
+DEAL:Cell-2d-Hermite-3::3.40632167 0.01840677
+DEAL:Cell-2d-Hermite-3::3.42545831 0.02731473
+DEAL:Cell-2d-Hermite-3::3.44459495 0.03780628
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+DEAL:Cell-2d-Hermite-3::3.48286823 0.04821598
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+DEAL:Cell-2d-Hermite-3::3.57855142 0.02304228
+DEAL:Cell-2d-Hermite-3::3.59768806 0.02689774
+DEAL:Cell-2d-Hermite-3::3.61682470 0.03193816
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+DEAL:Cell-2d-Hermite-3::3.75078117 0.03546890
+DEAL:Cell-2d-Hermite-3::3.76991781 0.03527373
+DEAL:Cell-2d-Hermite-3::3.78905445 0.03528177
+DEAL:Cell-2d-Hermite-3::3.80819108 0.03532578
+DEAL:Cell-2d-Hermite-3::3.82732772 0.03527734
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+DEAL:Cell-2d-Hermite-3::3.86560100 0.03497344
+DEAL:Cell-2d-Hermite-3::3.88473764 0.03496174
+DEAL:Cell-2d-Hermite-3::3.90387428 0.03513947
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+DEAL:Cell-2d-Hermite-3::3.94214755 0.03562431
+DEAL:Cell-2d-Hermite-3::3.96128419 0.03564035
+DEAL:Cell-2d-Hermite-3::3.98042083 0.03547483
+DEAL:Cell-2d-Hermite-3::3.99955747 0.03526028
+DEAL:Cell-2d-Hermite-3::4.01869411 0.03515019
+
+
projects / dealii.git / commitdiff