From 29ff7c34207802ed46996b681703c15470249eef Mon Sep 17 00:00:00 2001 From: Ivy Weber Date: Thu, 17 Mar 2022 11:31:54 +0100 Subject: [PATCH] Added files for implementing Hermite finite elements, and test files to ensure correct function. --- doc/doxygen/references.bib | 9 + include/deal.II/base/polynomials_hermite.h | 158 +++ include/deal.II/fe/fe_hermite.h | 248 ++++ .../matrix_free/shape_info.templates.h | 5 +- .../numerics/matrix_creator.templates.h | 14 +- include/deal.II/numerics/vector_tools.h | 1 + .../deal.II/numerics/vector_tools.templates.h | 1 + .../deal.II/numerics/vector_tools_hermite.h | 133 ++ .../numerics/vector_tools_hermite.templates.h | 649 ++++++++++ source/base/CMakeLists.txt | 1 + source/base/polynomials_hermite.cc | 122 ++ source/dofs/dof_tools_sparsity.cc | 27 +- source/fe/CMakeLists.txt | 2 + source/fe/fe_hermite.cc | 802 ++++++++++++ source/fe/fe_hermite.inst.in | 55 + source/numerics/CMakeLists.txt | 2 + source/numerics/vector_tools_hermite.cc | 23 + source/numerics/vector_tools_hermite.inst.in | 68 + tests/fe/2d_grid_projection_hermite.cc | 223 ++++ tests/fe/2d_grid_projection_hermite.output | 25 + tests/fe/derivatives.h | 172 +++ tests/fe/derivatives_hermite.cc | 78 ++ tests/fe/derivatives_hermite.output | 206 +++ tests/fe/shapes.h | 4 + tests/fe/shapes_hermite.cc | 87 ++ tests/fe/shapes_hermite.output | 1102 +++++++++++++++++ tests/fe/uneven_1d_grid_hermite.cc | 192 +++ tests/fe/uneven_1d_grid_hermite.output | 53 + tests/fe/uneven_1d_grid_hermite_laplace.cc | 250 ++++ .../fe/uneven_1d_grid_hermite_laplace.output | 57 + tests/numerics/project_boundaries_hermite.cc | 275 ++++ .../project_boundaries_hermite.output | 81 ++ tests/numerics/wave_solver_hermite.cc | 423 +++++++ tests/numerics/wave_solver_hermite.output | 963 ++++++++++++++ 34 files changed, 6504 insertions(+), 7 deletions(-) create mode 100644 include/deal.II/base/polynomials_hermite.h create mode 100644 include/deal.II/fe/fe_hermite.h create mode 100644 include/deal.II/numerics/vector_tools_hermite.h create mode 100644 include/deal.II/numerics/vector_tools_hermite.templates.h create mode 100644 source/base/polynomials_hermite.cc create mode 100644 source/fe/fe_hermite.cc create mode 100644 source/fe/fe_hermite.inst.in create mode 100644 source/numerics/vector_tools_hermite.cc create mode 100644 source/numerics/vector_tools_hermite.inst.in create mode 100644 tests/fe/2d_grid_projection_hermite.cc create mode 100644 tests/fe/2d_grid_projection_hermite.output create mode 100644 tests/fe/derivatives.h create mode 100644 tests/fe/derivatives_hermite.cc create mode 100644 tests/fe/derivatives_hermite.output create mode 100644 tests/fe/shapes_hermite.cc create mode 100644 tests/fe/shapes_hermite.output create mode 100644 tests/fe/uneven_1d_grid_hermite.cc create mode 100644 tests/fe/uneven_1d_grid_hermite.output create mode 100644 tests/fe/uneven_1d_grid_hermite_laplace.cc create mode 100644 tests/fe/uneven_1d_grid_hermite_laplace.output create mode 100644 tests/numerics/project_boundaries_hermite.cc create mode 100644 tests/numerics/project_boundaries_hermite.output create mode 100644 tests/numerics/wave_solver_hermite.cc create mode 100644 tests/numerics/wave_solver_hermite.output diff --git a/doc/doxygen/references.bib b/doc/doxygen/references.bib index ca27997f69..f39962e0d0 100644 --- a/doc/doxygen/references.bib +++ b/doc/doxygen/references.bib @@ -1774,3 +1774,12 @@ url = {https://doi.org/10.1016/0045-7930(73)90027-3} 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} +} diff --git a/include/deal.II/base/polynomials_hermite.h b/include/deal.II/base/polynomials_hermite.h new file mode 100644 index 0000000000..e19ef5f304 --- /dev/null +++ b/include/deal.II/base/polynomials_hermite.h @@ -0,0 +1,158 @@ +// --------------------------------------------------------------------- +// +// 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 + +#include +#include +#include +#include + +#include + +#include +#include +#include +#include + + + +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 + { + 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> + 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 side. + */ + 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 diff --git a/include/deal.II/fe/fe_hermite.h b/include/deal.II/fe/fe_hermite.h new file mode 100644 index 0000000000..628b9e4171 --- /dev/null +++ b/include/deal.II/fe/fe_hermite.h @@ -0,0 +1,248 @@ +// --------------------------------------------------------------------- +// +// 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 +#include + +#include + +#include +#include + +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 $(2r+2)^{\mathtt{dim} }-1$: + * + * FE_Hermite<1>(1) + * + * @verbatim + * (0)________________(2) + * (1) (3) + * @endverbatim + * + * FE_Hermite<2>(1): + * + * @verbatim + * ( 8, 9)__________(10,11) + * (12,13) (14,15) + * | | + * | | + * | | + * | | + * | | + * | | + * ( 0, 1)__________( 2, 3) + * ( 4, 5) ( 6, 7) + * @endverbatim + * + * FE_Hermite<3>(1): + * + * @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 +class FE_Hermite : public FE_Poly +{ +public: + /** + * Constructor that creates an Hermite finite element that imposes + * continuity in derivatives up to and including order @p regularity. + */ + FE_Hermite(const unsigned int regularity); + + // Other functions + /** + * Returns FE_Hermite(regularity) as a + * std::string 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> + 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 + 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::cell_iterator &cell, + const CellSimilarity::Similarity cell_similarity, + const Quadrature & quadrature, + const Mapping & mapping, + const typename Mapping::InternalDataBase &mapping_internal, + const dealii::internal::FEValuesImplementation::MappingRelatedData + & mapping_data, + const typename FiniteElement::InternalDataBase &fe_internal, + dealii::internal::FEValuesImplementation::FiniteElementRelatedData + &output_data) const override; + + using FiniteElement::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::cell_iterator &cell, + const unsigned int face_no, + const hp::QCollection & quadrature, + const Mapping & mapping, + const typename Mapping::InternalDataBase &mapping_internal, + const dealii::internal::FEValuesImplementation::MappingRelatedData + & mapping_data, + const typename FiniteElement::InternalDataBase &fe_internal, + dealii::internal::FEValuesImplementation::FiniteElementRelatedData + &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::InternalDataBase> + get_data( + const UpdateFlags update_flags, + const Mapping &mapping, + const Quadrature & quadrature, + dealii::internal::FEValuesImplementation::FiniteElementRelatedData + &output_data) const override + { + std::unique_ptr::InternalDataBase> + data_ptr = FE_Poly::get_data(update_flags, + mapping, + quadrature, + output_data); + auto &data = + dynamic_cast::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 diff --git a/include/deal.II/matrix_free/shape_info.templates.h b/include/deal.II/matrix_free/shape_info.templates.h index 9f1a084a81..9aa79ef071 100644 --- a/include/deal.II/matrix_free/shape_info.templates.h +++ b/include/deal.II/matrix_free/shape_info.templates.h @@ -29,6 +29,7 @@ #include #include #include +#include #include #include #include @@ -1095,7 +1096,9 @@ namespace internal return false; // then check if the base element is supported or not - if (dynamic_cast *>(fe_ptr) != nullptr) + if (dynamic_cast *>(fe_ptr) != nullptr && + dynamic_cast *>(fe_ptr) == + nullptr) { const FE_Poly *fe_poly_ptr = dynamic_cast *>(fe_ptr); diff --git a/include/deal.II/numerics/matrix_creator.templates.h b/include/deal.II/numerics/matrix_creator.templates.h index 90f008e0c4..76343bd922 100644 --- a/include/deal.II/numerics/matrix_creator.templates.h +++ b/include/deal.II/numerics/matrix_creator.templates.h @@ -28,6 +28,7 @@ #include #include +#include #include #include @@ -1001,7 +1002,9 @@ namespace MatrixCreator 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 *>(&fe) != nullptr); const bool fe_is_primitive = fe.is_primitive(); copy_data.cell = cell; @@ -1010,6 +1013,8 @@ namespace MatrixCreator 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 fe_values(mapping, fe, q, update_flags); // two variables for the coefficient, one for the two cases @@ -1353,9 +1358,14 @@ namespace MatrixCreator const FiniteElement &fe = dof.get_fe(); const unsigned int n_components = fe.n_components(); + const bool fe_is_hermite = + (dynamic_cast *>(&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()); diff --git a/include/deal.II/numerics/vector_tools.h b/include/deal.II/numerics/vector_tools.h index f3624336cd..6912fa99b2 100644 --- a/include/deal.II/numerics/vector_tools.h +++ b/include/deal.II/numerics/vector_tools.h @@ -23,6 +23,7 @@ #include #include #include +#include #include #include #include diff --git a/include/deal.II/numerics/vector_tools.templates.h b/include/deal.II/numerics/vector_tools.templates.h index 878b8fe7d4..0de1aa99b3 100644 --- a/include/deal.II/numerics/vector_tools.templates.h +++ b/include/deal.II/numerics/vector_tools.templates.h @@ -19,6 +19,7 @@ #include #include +#include #include #include #include diff --git a/include/deal.II/numerics/vector_tools_hermite.h b/include/deal.II/numerics/vector_tools_hermite.h new file mode 100644 index 0000000000..66efab2167 --- /dev/null +++ b/include/deal.II/numerics/vector_tools_hermite.h @@ -0,0 +1,133 @@ +// --------------------------------------------------------------------- +// +// 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 + +#include +#include +#include +#include + +#include + +#include + +#include +#include + +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 + void + project_hermite_boundary_values( + const Mapping & mapping_h, + const DoFHandler &dof_handler, + const std::map *> + & boundary_functions, + const Quadrature & quadrature, + const HermiteBoundaryType projection_mode, + std::map &boundary_values, + std::vector component_mapping = {}); + + + + /** + * The following function is a specialisation of the above for the case where + * Dirichlet boundary conditions should be enforced. + */ + template + void + project_hermite_boundary_values( + const Mapping & mapping_h, + const DoFHandler &dof_handler, + const std::map *> + & boundary_functions, + const Quadrature & quadrature, + std::map &boundary_values, + std::vector 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 + void + project_hermite( + const Mapping & mapping, + const DoFHandler & dofhandler, + const AffineConstraints & constraints, + const Quadrature & quadrature, + const Function &function, + VectorType & vec, + const bool enforce_zero_boundary = false, + const Quadrature &q_boundary = (dim > 1 ? QGauss(2) : + Quadrature(0)), + const bool project_to_boundary_first = false); + + /** @} */ +} // namespace VectorTools + +DEAL_II_NAMESPACE_CLOSE + +#endif diff --git a/include/deal.II/numerics/vector_tools_hermite.templates.h b/include/deal.II/numerics/vector_tools_hermite.templates.h new file mode 100644 index 0000000000..56d2cfed57 --- /dev/null +++ b/include/deal.II/numerics/vector_tools_hermite.templates.h @@ -0,0 +1,649 @@ +// --------------------------------------------------------------------- +// +// 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 +#include +#include +#include +#include + +#include + +#include +#include +#include +#include +#include + +#include +#include +#include +#include +#include +#include +#include +#include +#include + +#include +#include +#include +#include +#include + + +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 +using MappingHermite = MappingCartesian; + +namespace VectorTools +{ + // Internal namespace for implement Hermite boundary projection methods + namespace internal + { + template + void + get_constrained_hermite_boundary_dofs( + const DoFHandler &dof_handler, + const std::map *> + & boundary_functions, + const unsigned int position, + std::vector &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 &>(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 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 l2h = + dynamic_cast &>(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 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 + void + do_hermite_direct_projection( + const Mapping & mapping_h, + const DoFHandler &dof_handler, + const std::map *> + & boundary_functions, + const Quadrature & quadrature, + const unsigned int position, + std::map &boundary_values, + std::vector 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 *>(&mapping_h) != + nullptr) || + (dynamic_cast *>(&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::face_indices()) + if (cell->at_boundary(direction) && + (boundary_functions.find( + cell->face(direction)->boundary_id()) != + boundary_functions.end())) + { + const Function ¤t_function = + *boundary_functions + .find(cell->face(direction)->boundary_id()) + ->second; + + const FiniteElement &fe_herm = + dof_handler.get_fe(); + + AssertDimension(fe_herm.n_components(), + current_function.n_components); + + Vector 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 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*>(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 mass_matrix(sparsity); + Vector rhs(sparsity.n_rows()); + Vector 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 *>(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 + void + project_hermite_boundary_values( + const Mapping & mapping_h, + const DoFHandler &dof_handler, + const std::map *> + & boundary_functions, + const Quadrature & quadrature, + const HermiteBoundaryType projection_mode, + std::map &boundary_values, + std::vector 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 *>(&mapping_h) != + nullptr) || + (dynamic_cast *>(&mapping_h) != + nullptr), + ExcMessage("project_hermite_boundary_values should only be" + " used with a mapping compatible with Hermite " + "finite element methods.")); + Assert((dynamic_cast *>(&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 + void + project_hermite_boundary_values( + const Mapping & mapping_h, + const DoFHandler &dof_handler, + const std::map *> + & boundary_functions, + const Quadrature & quadrature, + std::map &boundary_values, + std::vector component_mapping) + { + Assert((dynamic_cast *>(&mapping_h) != + nullptr) || + (dynamic_cast *>(&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 + void + project_hermite( + const Mapping & mapping_h, + const DoFHandler & dof, + const AffineConstraints & constraints, + const Quadrature & quadrature, + const Function &function, + VectorType & vec, + const bool enforce_zero_boundary, + const Quadrature &q_boundary, + const bool project_to_boundary_first) + { + using number = typename VectorType::value_type; + + Assert((dynamic_cast *>(&mapping_h) != + nullptr) || + (dynamic_cast *>(&mapping_h) != + nullptr), + ExcMessage( + "The function project_hermite() can only be used with a mapping " + "compatible with Hermite finite element methods.")); + + Assert((dynamic_cast *>(&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 boundary_values; + const std::vector active_boundary_ids = + dof.get_triangulation().get_boundary_ids(); + + if (enforce_zero_boundary) + { + std::map *> + boundary; + + for (const auto it : active_boundary_ids) + boundary.emplace(std::make_pair(it, nullptr)); + + const std::map *> + new_boundary = + static_cast *>>( + boundary); + + std::vector 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 *> + boundary_function; + + for (const auto it : active_boundary_ids) + boundary_function.emplace(std::make_pair(it, &function)); + + const std::map *> + new_boundary_2 = + static_cast *>>( + boundary_function); + + project_hermite_boundary_values( + 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 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 mass_matrix(sparsity); + Vector 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 *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 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::set(vec_result(i), i, vec); + } +} // namespace VectorTools + + + +DEAL_II_NAMESPACE_CLOSE + +#endif diff --git a/source/base/CMakeLists.txt b/source/base/CMakeLists.txt index 78190e094a..42bb1aa5aa 100644 --- a/source/base/CMakeLists.txt +++ b/source/base/CMakeLists.txt @@ -73,6 +73,7 @@ SET(_unity_include_src polynomials_bernstein.cc polynomials_bdm.cc polynomials_bernardi_raugel.cc + polynomials_hermite.cc polynomials_nedelec.cc polynomials_integrated_legendre_sz.cc polynomial_space.cc diff --git a/source/base/polynomials_hermite.cc b/source/base/polynomials_hermite.cc new file mode 100644 index 0000000000..c94ee9d25f --- /dev/null +++ b/source/base/polynomials_hermite.cc @@ -0,0 +1,122 @@ +// --------------------------------------------------------------------- +// +// 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_NAMESPACE_OPEN + +namespace Polynomials +{ + namespace + { + std::vector + 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(regularity + 1 - curr_index); + const int comparison_2 = side ? static_cast(curr_index + 1) : + static_cast(regularity + 2); + + std::vector 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(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(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(hermite_poly_coeffs(regularity, index)) + , degree(2 * regularity + 1) + , regularity(regularity) + , side_index(index % (regularity + 1)) + , side((index > regularity) ? 1 : 0) + {} + + + + std::vector> + HermitePoly::generate_complete_basis(const unsigned int regularity) + { + std::vector> 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 diff --git a/source/dofs/dof_tools_sparsity.cc b/source/dofs/dof_tools_sparsity.cc index 83b90379c1..b447183cdc 100644 --- a/source/dofs/dof_tools_sparsity.cc +++ b/source/dofs/dof_tools_sparsity.cc @@ -26,6 +26,7 @@ #include #include +#include #include #include @@ -471,12 +472,20 @@ namespace DoFTools Assert(boundary_ids.find(numbers::internal_face_boundary_id) == boundary_ids.end(), (typename DoFHandler::ExcInvalidBoundaryIndicator())); - Assert(sparsity.n_rows() == dof.n_boundary_dofs(boundary_ids), + + const bool fe_is_hermite = (dynamic_cast *>( + &(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) { @@ -502,10 +511,20 @@ namespace DoFTools 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]]); } } diff --git a/source/fe/CMakeLists.txt b/source/fe/CMakeLists.txt index c335b309e5..0cbc4242e1 100644 --- a/source/fe/CMakeLists.txt +++ b/source/fe/CMakeLists.txt @@ -31,6 +31,7 @@ SET(_unity_include_src fe_dg_vector.cc fe_enriched.cc fe_face.cc + fe_hermite.cc fe_nedelec.cc fe_nedelec_sz.cc fe_nothing.cc @@ -103,6 +104,7 @@ SET(_inst 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 diff --git a/source/fe/fe_hermite.cc b/source/fe/fe_hermite.cc new file mode 100644 index 0000000000..328eb51107 --- /dev/null +++ b/source/fe/fe_hermite.cc @@ -0,0 +1,802 @@ +// --------------------------------------------------------------------- +// +// 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 +#include +#include +#include + +#include + +#include +#include +#include +#include +#include + +#include +#include +#include +#include +#include +#include +#include +#include +#include + +#include + +#include +#include +#include +#include +#include +#include + +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 +using MappingHermite = MappingCartesian; + + + +namespace internal +{ + inline std::vector + get_hermite_dpo_vector(const unsigned int dim, const unsigned int regularity) + { + std::vector 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 + void + hermite_hierarchic_to_lexicographic_numbering(const unsigned int regularity, + std::vector &h2l); + + + + template <> + void + hermite_hierarchic_to_lexicographic_numbering<1>( + const unsigned int regularity, + std::vector &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 &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 &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 + inline std::vector + hermite_hierarchic_to_lexicographic_numbering(const unsigned int regularity) + { + const std::vector dpo = + get_hermite_dpo_vector(dim, regularity); + const dealii::FiniteElementData face_data(dpo, 1, 2 * regularity + 1); + std::vector renumbering(face_data.dofs_per_cell); + + hermite_hierarchic_to_lexicographic_numbering(regularity, renumbering); + + return renumbering; + } + + + + template + std::vector + hermite_lexicographic_to_hierarchic_numbering(const unsigned int regularity) + { + return Utilities::invert_permutation( + hermite_hierarchic_to_lexicographic_numbering(regularity)); + } + + + + template + inline std::vector + hermite_face_lexicographic_to_hierarchic_numbering( + const unsigned int regularity) + { + return (dim > 1) ? hermite_lexicographic_to_hierarchic_numbering( + regularity) : + std::vector(); + } + + + + template + TensorProductPolynomials + get_hermite_polynomials(const unsigned int regularity) + { + TensorProductPolynomials poly_space( + Polynomials::HermitePoly::generate_complete_basis(regularity)); + + std::vector renumber = + internal::hermite_hierarchic_to_lexicographic_numbering(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 + class Rescaler + { + public: + void + rescale_fe_hermite_values( + const FE_Hermite & fe_herm, + const typename Mapping::InternalDataBase &mapping_data, + Table<2, xNumber> & value_list); + + + + template + 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::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::InternalData *>( + &mapping_data) != nullptr) + { + const typename MappingCartesian<1>::InternalData *data = + dynamic_cast::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 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 + 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::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::InternalData *>( + &mapping_data) != nullptr) + { + const typename MappingCartesian<2>::InternalData *data = + dynamic_cast::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 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 + 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::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::InternalData *>( + &mapping_data) != nullptr) + { + const typename MappingCartesian<3>::InternalData *data = + dynamic_cast::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 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 +FE_Hermite::FE_Hermite(const unsigned int regularity) + : FE_Poly( + internal::get_hermite_polynomials(regularity), + FiniteElementData(internal::get_hermite_dpo_vector(dim, regularity), + 1, + 2 * regularity + 1, + (regularity > 0 ? FiniteElementData::H2 : + + FiniteElementData::H1)), + std::vector(Utilities::pow(2 * (regularity + 1), dim), false), + std::vector(Utilities::pow(2 * (regularity + 1), dim), + std::vector(1, true))) + , regularity(regularity) +{} + + + +template +std::string +FE_Hermite::get_name() const +{ + std::ostringstream name_buffer; + name_buffer << "FE_Hermite<" << dim << "," << spacedim << ">(" + << this->regularity << ")"; + return name_buffer.str(); +} + + + +template +std::unique_ptr> +FE_Hermite::clone() const +{ + return std::make_unique>(*this); +} + + + +template +std::vector +FE_Hermite::get_lexicographic_to_hierarchic_numbering() const +{ + return internal::hermite_lexicographic_to_hierarchic_numbering( + this->regularity); +} + + + +template +void +FE_Hermite::fill_fe_values( + const typename Triangulation::cell_iterator &, + const CellSimilarity::Similarity cell_similarity, + const Quadrature & /*quadrature*/, + const Mapping & mapping, + const typename Mapping::InternalDataBase &mapping_internal, + const dealii::internal::FEValuesImplementation::MappingRelatedData + & /*mapping_data*/, + const typename FiniteElement::InternalDataBase &fe_internal, + dealii::internal::FEValuesImplementation::FiniteElementRelatedData + &output_data) const +{ + // Convert data object to internal data for this class. + // Fails with an exception if that is not possible. + Assert( + (dynamic_cast::InternalData *>( + &fe_internal) != nullptr), + ExcInternalError()); + const typename FE_Hermite::InternalData &fe_data = + static_cast::InternalData &>( + fe_internal); + + Assert( + (dynamic_cast::InternalData *>( + &mapping_internal) != nullptr) || + (dynamic_cast::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 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> 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> 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> third_dev_fix; + third_dev_fix.rescale_fe_hermite_values( + third_dev_fix, + *this, + mapping_internal, + output_data.shape_3rd_derivatives); + } +} + + + +template +void +FE_Hermite::fill_fe_face_values( + const typename Triangulation::cell_iterator &cell, + const unsigned int face_no, + const hp::QCollection & quadrature, + const Mapping & mapping, + const typename Mapping::InternalDataBase & mapping_internal, + const dealii::internal::FEValuesImplementation::MappingRelatedData + &, + const typename FiniteElement::InternalDataBase &fe_internal, + dealii::internal::FEValuesImplementation::FiniteElementRelatedData + &output_data) const +{ + /* + * Convert data object to internal data for this class. Fails with + * an exception if that is not possible. + */ + Assert( + (dynamic_cast::InternalData *>( + &fe_internal) != nullptr), + ExcInternalError()); + const typename FE_Hermite::InternalData &fe_data = + static_cast::InternalData &>( + fe_internal); + + Assert( + (dynamic_cast::InternalData *>( + &mapping_internal) != nullptr) || + (dynamic_cast::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::DataSetDescriptor offset = + QProjector::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 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> 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> 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> 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 diff --git a/source/fe/fe_hermite.inst.in b/source/fe/fe_hermite.inst.in new file mode 100644 index 0000000000..bfc973b766 --- /dev/null +++ b/source/fe/fe_hermite.inst.in @@ -0,0 +1,55 @@ +// --------------------------------------------------------------------- +// +// 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( + const unsigned int regularity, std::vector &h2l); + + template std::vector + internal::hermite_lexicographic_to_hierarchic_numbering( + const unsigned int regularity); + + template std::vector internal:: + hermite_face_lexicographic_to_hierarchic_numbering( + 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>; +#endif + } + + + +for (deal_II_dimension : DIMENSIONS; deal_II_space_dimension : SPACE_DIMENSIONS) + { +#if deal_II_dimension <= deal_II_space_dimension + template class internal::Rescaler; + template class FE_Hermite; +#endif + } diff --git a/source/numerics/CMakeLists.txt b/source/numerics/CMakeLists.txt index 9b0031d40a..563b3e1995 100644 --- a/source/numerics/CMakeLists.txt +++ b/source/numerics/CMakeLists.txt @@ -53,6 +53,7 @@ SET(_separate_src 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 @@ -95,6 +96,7 @@ SET(_inst 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 diff --git a/source/numerics/vector_tools_hermite.cc b/source/numerics/vector_tools_hermite.cc new file mode 100644 index 0000000000..419e2de208 --- /dev/null +++ b/source/numerics/vector_tools_hermite.cc @@ -0,0 +1,23 @@ +// --------------------------------------------------------------------- +// +// 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_NAMESPACE_OPEN + +// explicit instantiations +#include "vector_tools_hermite.inst" + +DEAL_II_NAMESPACE_CLOSE diff --git a/source/numerics/vector_tools_hermite.inst.in b/source/numerics/vector_tools_hermite.inst.in new file mode 100644 index 0000000000..ede0c3a233 --- /dev/null +++ b/source/numerics/vector_tools_hermite.inst.in @@ -0,0 +1,68 @@ +// --------------------------------------------------------------------- +// +// 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 & mapping_h, + const DoFHandler &dof_handler, + const std::map *> + & boundary_functions, + const Quadrature & quadrature, + const unsigned int position, + std::map &boundary_values, + std::vector component_mapping); + + template void VectorTools::project_hermite_boundary_values< + deal_II_dimension, + deal_II_dimension, + double>(const Mapping & mapping_h, + const DoFHandler &dof_handler, + const std::map *> + & boundary_functions, + const Quadrature & quadrature, + const HermiteBoundaryType projection_mode, + std::map &boundary_values, + std::vector component_mapping); + + template void VectorTools::project_hermite_boundary_values< + deal_II_dimension, + deal_II_dimension, + double>(const Mapping & mapping_h, + const DoFHandler &dof_handler, + const std::map *> + & boundary_functions, + const Quadrature & quadrature, + std::map &boundary_values, + std::vector component_mapping); + + template void VectorTools:: + project_hermite, deal_II_dimension>( + const Mapping & mapping, + const DoFHandler & dof, + const AffineConstraints::value_type> &constraints, + const Quadrature & quadrature, + const Function::value_type> &function, + Vector & vec, + const bool enforce_zero_boundary, + const Quadrature &q_boundary, + const bool project_to_boundary_first); + } diff --git a/tests/fe/2d_grid_projection_hermite.cc b/tests/fe/2d_grid_projection_hermite.cc new file mode 100644 index 0000000000..cd1ba984be --- /dev/null +++ b/tests/fe/2d_grid_projection_hermite.cc @@ -0,0 +1,223 @@ +// --------------------------------------------------------------------- +// +// 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 + +#include +#include +#include + +#include +#include +#include + +#include +#include +#include +#include + +#include +#include +#include +#include + +#include +#include +#include +#include +#include +#include +#include + +#include +#include +#include + +#include +#include +#include +#include +#include +#include + + + +/* + * Test case for Hermite on an irregular 1D grid. + * FE_Hermite(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 +class Solution : public Function +{ +public: + virtual double + value(const Point &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 +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 tr; + DoFHandler dof(tr); + + double left = -1.0, right = 1.0; + Point 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 herm(regularity); + dof.distribute_dofs(herm); + + MappingCartesian mapping; + + QGauss quadr(2 * regularity + 2); + + Vector sol(dof.n_dofs()); + Vector rhs(dof.n_dofs()); + + Solution sol_object; + + AffineConstraints constraints; + constraints.close(); + + DynamicSparsityPattern dsp(dof.n_dofs()); + DoFTools::make_sparsity_pattern(dof, dsp); + SparsityPattern sp; + sp.copy_from(dsp); + + SparseMatrix mass_matrix; + mass_matrix.reinit(sp); + MatrixCreator::create_mass_matrix(mapping, dof, quadr, mass_matrix); + + FEValues fe_herm(mapping, + herm, + quadr, + update_values | update_quadrature_points | + update_JxW_values); + + std::vector 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; +} diff --git a/tests/fe/2d_grid_projection_hermite.output b/tests/fe/2d_grid_projection_hermite.output new file mode 100644 index 0000000000..a6bc9d65e6 --- /dev/null +++ b/tests/fe/2d_grid_projection_hermite.output @@ -0,0 +1,25 @@ + +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 + + diff --git a/tests/fe/derivatives.h b/tests/fe/derivatives.h new file mode 100644 index 0000000000..917a48afa8 --- /dev/null +++ b/tests/fe/derivatives.h @@ -0,0 +1,172 @@ +// --------------------------------------------------------------------- +// +// 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 + +#include + +#include +#include + +#include +#include +#include + +#include + +#include +#include +#include +#include + +#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 +inline void +plot_function_derivatives(Mapping & mapping, + FiniteElement &finel, + const char * name) +{ + Triangulation tr; + DoFHandler dof(tr); + GridGenerator::hyper_cube(tr, 0., 1.); + dof.distribute_dofs(finel); + + typename DoFHandler::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(degree - 1) / 2); + const unsigned int div = 1; + + QTrapez q; + FEValues 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 diff --git a/tests/fe/derivatives_hermite.cc b/tests/fe/derivatives_hermite.cc new file mode 100644 index 0000000000..f450361965 --- /dev/null +++ b/tests/fe/derivatives_hermite.cc @@ -0,0 +1,78 @@ +// --------------------------------------------------------------------- +// +// 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 +#include + +#include + +#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 +void +print_hermite_endpoint_derivatives() +{ + MappingCartesian m; + + FE_Hermite herm0(0); + plot_function_derivatives(m, herm0, "Hermite-0"); + + FE_Hermite herm1(1); + plot_function_derivatives(m, herm1, "Hermite-1"); + + // Skip the following for dim 3 or greater + if (dim < 3) + { + FE_Hermite herm2(2); + plot_function_derivatives(m, herm2, "Hermite-2"); + } + if (dim == 1) + { + FE_Hermite herm3(3); + plot_function_derivatives(m, herm3, "Hermite-3"); + + FE_Hermite herm4(4); + plot_function_derivatives(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; +} diff --git a/tests/fe/derivatives_hermite.output b/tests/fe/derivatives_hermite.output new file mode 100644 index 0000000000..e3a15195df --- /dev/null +++ b/tests/fe/derivatives_hermite.output @@ -0,0 +1,206 @@ + +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 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1.00000000 1.00000000 1.00000000 65.00000000 +DEAL:Cell-3d-Hermite-1:: +DEAL:Cell-3d-Hermite-1:: +DEAL:Cell-3d-Hermite-1:: diff --git a/tests/fe/shapes.h b/tests/fe/shapes.h index b7de15072b..72a6cc7706 100644 --- a/tests/fe/shapes.h +++ b/tests/fe/shapes.h @@ -13,6 +13,8 @@ // // --------------------------------------------------------------------- +#ifndef SHAPES_TEST_HEADER +#define SHAPES_TEST_HEADER // Show the shape functions implemented. @@ -515,3 +517,5 @@ test_compute_functions(const Mapping & mapping, check_values_and_derivatives(fe, fe_values, q); }; } + +#endif diff --git a/tests/fe/shapes_hermite.cc b/tests/fe/shapes_hermite.cc new file mode 100644 index 0000000000..56877045dd --- /dev/null +++ b/tests/fe/shapes_hermite.cc @@ -0,0 +1,87 @@ +// --------------------------------------------------------------------- +// +// 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 +#include + +#include + +#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 +void +plot_FE_Hermite_shape_functions() +{ + MappingCartesian m; + + FE_Hermite 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 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 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 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 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; +} diff --git a/tests/fe/shapes_hermite.output b/tests/fe/shapes_hermite.output new file mode 100644 index 0000000000..c22d1751d8 --- /dev/null +++ b/tests/fe/shapes_hermite.output @@ -0,0 +1,1102 @@ + +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:: 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1.00000000 1.00000000 +DEAL:Face3d-Hermite-1:: +DEAL:Face3d-Hermite-1:: diff --git a/tests/fe/uneven_1d_grid_hermite.cc b/tests/fe/uneven_1d_grid_hermite.cc new file mode 100644 index 0000000000..27f27189f8 --- /dev/null +++ b/tests/fe/uneven_1d_grid_hermite.cc @@ -0,0 +1,192 @@ +// --------------------------------------------------------------------- +// +// 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 +#include +#include + +#include +#include +#include + +#include +#include +#include +#include + +#include +#include +#include +#include + +#include +#include +#include +#include +#include +#include +#include + +#include +#include +#include + +#include +#include +#include +#include +#include +#include + + + +/* + * Test case for Hermite on an irregular 1D grid. + * 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 solution(dof.n_dofs()); + TestPoly rhs_func; + + AffineConstraints constraints; + constraints.close(); + + FEValues<1> fe_herm(mapping, + herm, + quadr, + update_values | update_quadrature_points | + update_JxW_values); + + std::vector 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; +} diff --git a/tests/fe/uneven_1d_grid_hermite.output b/tests/fe/uneven_1d_grid_hermite.output new file mode 100644 index 0000000000..12b5d6c3d2 --- /dev/null +++ b/tests/fe/uneven_1d_grid_hermite.output @@ -0,0 +1,53 @@ + +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 + + diff --git a/tests/fe/uneven_1d_grid_hermite_laplace.cc b/tests/fe/uneven_1d_grid_hermite_laplace.cc new file mode 100644 index 0000000000..b53f510c33 --- /dev/null +++ b/tests/fe/uneven_1d_grid_hermite_laplace.cc @@ -0,0 +1,250 @@ +// --------------------------------------------------------------------- +// +// 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 +#include +#include + +#include +#include +#include + +#include +#include +#include +#include + +#include +#include +#include +#include + +#include +#include +#include +#include +#include +#include +#include + +#include +#include + +#include +#include +#include +#include +#include +#include + + + +/* + * Test case for Hermite on an irregular 1D grid. + * 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 sol(dof.n_dofs()); + Vector rhs(dof.n_dofs()); + + Solution sol_object; + RHSFunction rhs_object; + + AffineConstraints constraints; + constraints.close(); + + DynamicSparsityPattern dsp(dof.n_dofs()); + DoFTools::make_sparsity_pattern(dof, dsp); + SparsityPattern sp; + sp.copy_from(dsp); + + SparseMatrix 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 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 bound_vals; + + std::map *> 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; +} diff --git a/tests/fe/uneven_1d_grid_hermite_laplace.output b/tests/fe/uneven_1d_grid_hermite_laplace.output new file mode 100644 index 0000000000..1458fa5c78 --- /dev/null +++ b/tests/fe/uneven_1d_grid_hermite_laplace.output @@ -0,0 +1,57 @@ + +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 + + diff --git a/tests/numerics/project_boundaries_hermite.cc b/tests/numerics/project_boundaries_hermite.cc new file mode 100644 index 0000000000..9a0b6c50c5 --- /dev/null +++ b/tests/numerics/project_boundaries_hermite.cc @@ -0,0 +1,275 @@ +// --------------------------------------------------------------------- +// +// 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 + +#include +#include +#include + +#include +#include +#include + +#include +#include +#include +#include + +#include +#include +#include +#include + +#include +#include +#include +#include +#include +#include +#include + +#include +#include +#include + +#include +#include +#include +#include +#include +#include + + + +/* + * Test case for Hermite on an irregular 1D grid. + * FE_Hermite(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 +class Solution : public Function +{ +public: + virtual double + value(const Point &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 +class RHSFunction : public Function +{ +public: + virtual double + value(const Point &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 +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 tr; + DoFHandler dof(tr); + + double left = -1.0, right = 1.0; + GridGenerator::subdivided_hyper_cube(tr, 4, left, right); + + FE_Hermite herm(regularity); + dof.distribute_dofs(herm); + + MappingCartesian mapping; + + QGauss quadr(2 * regularity + 2); + + Vector sol(dof.n_dofs()); + Vector rhs(dof.n_dofs()); + + Solution sol_object; + RHSFunction rhs_object; + + AffineConstraints constraints; + constraints.close(); + + DynamicSparsityPattern dsp(dof.n_dofs()); + DoFTools::make_sparsity_pattern(dof, dsp); + SparsityPattern sp; + sp.copy_from(dsp); + + SparseMatrix stiffness_matrix; + stiffness_matrix.reinit(sp); + MatrixCreator::create_laplace_matrix(mapping, dof, quadr, stiffness_matrix); + + FEValues fe_herm(mapping, + herm, + quadr, + update_values | update_gradients | + update_quadrature_points | update_JxW_values); + + std::vector 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 bound_vals; + + std::map *> 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(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; +} diff --git a/tests/numerics/project_boundaries_hermite.output b/tests/numerics/project_boundaries_hermite.output new file mode 100644 index 0000000000..12b9a73158 --- /dev/null +++ b/tests/numerics/project_boundaries_hermite.output @@ -0,0 +1,81 @@ + +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 + + diff --git a/tests/numerics/wave_solver_hermite.cc b/tests/numerics/wave_solver_hermite.cc new file mode 100644 index 0000000000..7d8792184a --- /dev/null +++ b/tests/numerics/wave_solver_hermite.cc @@ -0,0 +1,423 @@ +// --------------------------------------------------------------------- +// +// 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 + +#include +#include +#include + +#include +#include +#include + +#include +#include +#include +#include + +#include +#include +#include +#include + +#include +#include +#include +#include +#include + +#include +#include +#include + +#include +#include +#include +#include +#include + + + +/* + * 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 +class Solution : public Function +{ +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 &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 +class SolutionDerivative : public Function +{ +public: + SolutionDerivative() = delete; + SolutionDerivative(const Solution &parent) + : parent_solution(parent){}; + + + + virtual double + value(const Point &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 parent_solution; +}; + + + +template +double +get_cfl_number(const unsigned int regularity) +{ + double c_eff = 1.0; + return c_eff / (std::sqrt(12.0 * dim) * (regularity + 1)); +} + + + +template +double +l2_error(const Mapping & mapping, + const DoFHandler &dof, + const Quadrature &quadrature, + const Function & solution, + const Vector & fe_solution) +{ + FEValues herm_vals(mapping, + dof.get_fe(), + quadrature, + update_values | update_quadrature_points | + update_JxW_values); + std::vector 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 +void +test_wave_solver(const double initial_time, const unsigned int regularity) +{ + Triangulation tr; + DoFHandler 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(regularity); + double final_time = 4.0; + + GridGenerator::subdivided_hyper_cube(tr, divisions, x_left, x_right); + + MappingCartesian mapping_h; + FE_Hermite fe_h(regularity); + dof.distribute_dofs(fe_h); + + AffineConstraints constraints; + constraints.close(); + + QGauss quadrature(2 * regularity + 2); + QGauss face_quadrature(2 * regularity + 2); + + Solution wave(initial_time); + SolutionDerivative wave_tdev(wave); + + Vector initial_conds_value(dof.n_dofs()); + Vector iniital_conds_tdev(dof.n_dofs()); + + Vector sol_prev(dof.n_dofs()); + Vector sol_curr(dof.n_dofs()); + Vector sol_next(dof.n_dofs()); + + Vector temp(dof.n_dofs()); + + SparsityPattern sp; + { + DynamicSparsityPattern dsp(dof.n_dofs()); + DoFTools::make_sparsity_pattern(dof, dsp); + sp.copy_from(dsp); + } + + SparseMatrix 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 boundary_values; + std::map *> boundary_functions; + + boundary_functions.emplace(std::make_pair(0U, &wave)); + if (dim == 1) + boundary_functions.emplace(std::make_pair(1U, &wave)); + + std::vector l2_errors; + double max_error; + l2_errors.emplace_back( + l2_error(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(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(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; +} diff --git a/tests/numerics/wave_solver_hermite.output b/tests/numerics/wave_solver_hermite.output new file mode 100644 index 0000000000..d7349f4994 --- /dev/null +++ b/tests/numerics/wave_solver_hermite.output @@ -0,0 +1,963 @@ + +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 +DEAL:Cell-2d-Hermite-3::2.77481260 0.02739423 +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 +DEAL:Cell-2d-Hermite-3::2.87049579 0.02850094 +DEAL:Cell-2d-Hermite-3::2.88963243 0.02868846 +DEAL:Cell-2d-Hermite-3::2.90876907 0.02897779 +DEAL:Cell-2d-Hermite-3::2.92790571 0.02928685 +DEAL:Cell-2d-Hermite-3::2.94704235 0.02954683 +DEAL:Cell-2d-Hermite-3::2.96617899 0.02976810 +DEAL:Cell-2d-Hermite-3::2.98531562 0.02999740 +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 +DEAL:Cell-2d-Hermite-3::3.10013546 0.03106542 +DEAL:Cell-2d-Hermite-3::3.11927209 0.03127554 +DEAL:Cell-2d-Hermite-3::3.13840873 0.03144977 +DEAL:Cell-2d-Hermite-3::3.15754537 0.03167178 +DEAL:Cell-2d-Hermite-3::3.17668201 0.03200583 +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 +DEAL:Cell-2d-Hermite-3::3.46373159 0.04537919 +DEAL:Cell-2d-Hermite-3::3.48286823 0.04821598 +DEAL:Cell-2d-Hermite-3::3.50200487 0.04599299 +DEAL:Cell-2d-Hermite-3::3.52114151 0.03963294 +DEAL:Cell-2d-Hermite-3::3.54027814 0.03126298 +DEAL:Cell-2d-Hermite-3::3.55941478 0.02439464 +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 +DEAL:Cell-2d-Hermite-3::3.63596134 0.03573738 +DEAL:Cell-2d-Hermite-3::3.65509798 0.03770588 +DEAL:Cell-2d-Hermite-3::3.67423461 0.03813368 +DEAL:Cell-2d-Hermite-3::3.69337125 0.03762268 +DEAL:Cell-2d-Hermite-3::3.71250789 0.03676313 +DEAL:Cell-2d-Hermite-3::3.73164453 0.03597467 +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 +DEAL:Cell-2d-Hermite-3::3.84646436 0.03512448 +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 +DEAL:Cell-2d-Hermite-3::3.92301092 0.03541644 +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 + + -- 2.39.5