Address = {University of Minnesota},
Number = {91-279},
Url = {citeseer.ist.psu.edu/saad93flexible.html}
+
+@article{ainsworth1998hp,
+ author = {Ainsworth, Mark and Senior, Bill},
+ title = {An adaptive refinement strategy for hp-finite element computations},
+ journal = {{Applied Numerical Mathematics}},
+ volume = {26},
+ number = {1--2},
+ pages = {165--178},
+ publisher = {Elsevier},
+ year = {1998},
+ doi = {10.1016/S0168-9274(97)00083-4}
+}
+
+@article{melenk2001hp,
+ author = {Melenk, Jens Markus and Wohlmuth, Barbara I.},
+ title = {{On residual-based a posteriori error estimation in hp-FEM}},
+ journal = {{Advances in Computational Mathematics}},
+ volume = {15},
+ number = {1},
+ pages = {311--331},
+ publisher = {Springer US},
+ year = {2001},
+ doi = {10.1023/A:1014268310921}
+}
+
+@article{mavriplis1994hp,
+ author = {Mavriplis, Catherine},
+ title = {Adaptive mesh strategies for the spectral element method},
+ journal = {{Computer Methods in Applied Mechanics and Engineering}},
+ year = {1994},
+ volume = {116},
+ number = {1},
+ pages = {77--86},
+ publisher = {Elsevier},
+ doi = {10.1016/S0045-7825(94)80010-3}
+}
+
+@article{houston2005hp,
+ author = {Houston, Paul and S{\"u}li, Endre},
+ title = {A note on the design of hp-adaptive finite element methods for elliptic partial differential equations},
+ journal = {{Computer Methods in Applied Mechanics and Engineering}},
+ number = {2},
+ pages = {229--243},
+ publisher = {Elsevier},
+ volume = {194},
+ year = {2005},
+ doi = {10.1016/j.cma.2004.04.009}
+}
+
+@article{eibner2007hp,
+ author = {Eibner, Tino and Melenk, Jens Markus},
+ title = {{An adaptive strategy for hp-FEM based on testing for analyticity}},
+ journal = {{Computational Mechanics}},
+ year = {2007},
+ volume = {39},
+ number = {5},
+ pages = {575--595},
+ publisher = {Springer},
+ doi = {10.1007/s00466-006-0107-0}
+}
+
+@article{davydov2017hp,
+ author = {Davydov, Denis and Gerasimov, Tymofiy and Pelteret, Jean-Paul and Steinmann, Paul},
+ title = {{Convergence study of the h-adaptive PUM and the hp-adaptive FEM applied to eigenvalue problems in quantum mechanics}},
+ journal = {{Advanced Modeling and Simulation in Engineering Sciences}},
+ year = {2017},
+ volume = {4},
+ number = {1},
+ pages = {7},
+ issn = {2213-7467},
+ doi = {10.1186/s40323-017-0093-0}
}
@Article{clevenger_par_gmg,
--- /dev/null
+Changed: FESeries::Fourier and FESeries::Legendre now require to use an
+individual number of coefficients per direction for each finite element
+registered in the provided FECollection.
+<br>
+(Marc Fehling, 2020/01/22)
--- /dev/null
+New: Namespace SmoothnessEstimator providing smoothness estimation strategies
+for hp-adaptive FEM based on Fourier and Legendre series expansions.
+<br>
+(Marc Fehling, Denis Davydov, 2019/10/30)
-New: Add FESeries::Legendre::get_size_in_each_direction() to retrieve
+New: Add FESeries::Legendre::get_n_coefficients_per_direction() and
+FESeries::Fourier::get_n_coefficients_per_direction() to retrieve
the number of coefficients in each direction. Also add an Assert in
-FESeries::Legendre::calculate() to check the dimension of the table to store
-coefficients.
+FESeries::Legendre::calculate() and FESeries::Fourier::calculate() to
+check the dimension of the table to store coefficients.
<br>
(Denis Davydov, 2018/12/27)
--- /dev/null
+New: Member function hp::FECollection::max_degree() that returns the
+maximal polynomial degree of all finite elements in the collection.
+<br>
+(Marc Fehling, 2019/10/30)
--- /dev/null
+New: Both classes FESeries::Fourier and FESeries::Legendre now offer a
+default constructor and can be initialized after their instantiation.
+<br>
+(Marc Fehling, 2020/01/21)
--- /dev/null
+New: FESeries::process_coefficients() now allows to ignore coefficients
+below an absolute threshold, which has to be provided as a parameter.
+<br>
+(Marc Fehling, 2020/01/21)
then consequently the function we had here was in $H^{\mu-d/2}$.
-<h4>What has to be done</h4>
+<h4>What we have to do</h4>
So what do we have to do to estimate the local smoothness of $u({\bf x})$ on
a cell $K$? Clearly, the first step is to compute the Fourier coefficients
@f]
with the matrix
@f[
- {\cal F}_{{\bf k},j}
- =
- \int_{\hat K} e^{i {\bf k}\cdot \hat{\bf x}} \hat \varphi_j(\hat{\bf x}) d\hat{\bf x}.
+ {\cal F}_{{\bf k},j}
+ =
+ \int_{\hat K} e^{i {\bf k}\cdot \hat{\bf x}} \hat \varphi_j(\hat{\bf x}) d\hat{\bf x}.
@f]
This matrix is easily computed for a given number of shape functions
$\varphi_j$ and Fourier modes $N$. Consequently, finding the
the exponent $\mu$ that we can then use to determine that
$\hat u(\hat{\bf x})$ is in $H^s(\hat K)$ with $s=\mu-\frac d2$.
-These steps outlined above are applicable on many different scenarios, which
-motivated the introduction of a generic function
-SmoothnessEstimator::estimate_by_coefficient_decay() in deal.II, that combines all
-the tasks described in this section in one simple function call.
<h4>Compensating for anisotropy</h4>
// These are the new files we need. The first and second provide <i>hp</i>
// versions of the DoFHandler and FEValues classes as described in the
-// introduction of this program. The last one provides the smoothness estimation
-// algorithms on decaying series expansion coefficients.
+// introduction of this program. The last one provides Fourier transformation
+// class on the unit cell.
#include <deal.II/hp/dof_handler.h>
#include <deal.II/hp/fe_values.h>
-#include <deal.II/numerics/smoothness_estimator.h>
+#include <deal.II/fe/fe_series.h>
// The last set of include files are standard C++ headers. We need support for
// complex numbers when we compute the Fourier transform.
void assemble_system();
void solve();
void create_coarse_grid();
+ void estimate_smoothness(Vector<float> &smoothness_indicators);
void postprocess(const unsigned int cycle);
+ std::pair<bool, unsigned int> predicate(const TableIndices<dim> &indices);
Triangulation<dim> triangulation;
hp::QCollection<dim> quadrature_collection;
hp::QCollection<dim - 1> face_quadrature_collection;
+ hp::QCollection<dim> fourier_q_collection;
+ FESeries::Fourier<dim> fourier;
+ std::vector<double> ln_k;
+ Table<dim, std::complex<double>> fourier_coefficients;
+
AffineConstraints<double> constraints;
SparsityPattern sparsity_pattern;
// face quadrature objects. We start with quadratic elements, and each
// quadrature formula is chosen so that it is appropriate for the matching
// finite element in the hp::FECollection object.
+ //
+ // Finally, we initialize FESeries::Fourier object which will be used to
+ // calculate coefficient in Fourier series as described in the introduction.
+ // In addition to the hp::FECollection, we need to provide quadrature rules
+ // hp::QCollection for integration on the reference cell.
+ //
+ // In order to resize fourier_coefficients Table, we use the following
+ // auxiliary function
+ template <int dim, typename T>
+ void resize(Table<dim, T> &coeff, const unsigned int N)
+ {
+ TableIndices<dim> size;
+ for (unsigned int d = 0; d < dim; d++)
+ size[d] = N;
+ coeff.reinit(size);
+ }
+
template <int dim>
LaplaceProblem<dim>::LaplaceProblem()
: dof_handler(triangulation)
quadrature_collection.push_back(QGauss<dim>(degree + 1));
face_quadrature_collection.push_back(QGauss<dim - 1>(degree + 1));
}
+
+ // As described in the introduction, we define the Fourier vectors ${\bf
+ // k}$ for which we want to compute Fourier coefficients of the solution
+ // on each cell as follows. In 2d, we will need coefficients corresponding
+ // to vectors ${\bf k}=(2 \pi i, 2\pi j)^T$ for which $\sqrt{i^2+j^2}\le N$,
+ // with $i,j$ integers and $N$ being the maximal polynomial degree we use
+ // for the finite elements in this program. The FESeries::Fourier class'
+ // constructor first parameter $N$ defines the number of coefficients in 1D
+ // with the total number of coefficients being $N^{dim}$. Although we will
+ // not use coefficients corresponding to
+ // $\sqrt{i^2+j^2}> N$ and $i+j==0$, the overhead of their calculation is
+ // minimal. The transformation matrices for each FiniteElement will be
+ // calculated only once the first time they are required in the course of
+ // hp-adaptive refinement. Because we work on the unit cell, we can do all
+ // this work without a mapping from reference to real cell and consequently
+ // can precalculate these matrices. The calculation of expansion
+ // coefficients for a particular set of local degrees of freedom on a given
+ // cell then follows as a simple matrix-vector product.
+ // The 3d case is handled analogously.
+ const unsigned int N = max_degree;
+
+ // We will need to assemble the matrices that do the Fourier transforms
+ // for each of the finite elements we deal with, i.e. the matrices ${\cal
+ // F}_{{\bf k},j}$ defined in the introduction. We have to do that for
+ // each of the finite elements in use. To that end we need a quadrature
+ // rule. In this example we use the same quadrature formula for each
+ // finite element, namely that is obtained by iterating a
+ // 2-point Gauss formula as many times as the maximal exponent we use for
+ // the term $e^{i{\bf k}\cdot{\bf x}}$:
+ QGauss<1> base_quadrature(2);
+ QIterated<dim> quadrature(base_quadrature, N);
+ for (unsigned int i = 0; i < fe_collection.size(); i++)
+ fourier_q_collection.push_back(quadrature);
+
+ // Now we are ready to set-up the FESeries::Fourier object
+ const std::vector<unsigned int> n_coefficients_per_direction(
+ fe_collection.size(), N);
+ fourier.initialize(n_coefficients_per_direction,
+ fe_collection,
+ fourier_q_collection);
+
+ // We need to resize the matrix of fourier coefficients according to the
+ // number of modes N.
+ resize(fourier_coefficients, N);
}
// Let us start with computing estimated error and smoothness indicators,
// which each are one number for each active cell of our
// triangulation. For the error indicator, we use the KellyErrorEstimator
- // class as always.
+ // class as always. Estimating the smoothness is done in the respective
+ // function of this class; that function is discussed further down below:
Vector<float> estimated_error_per_cell(triangulation.n_active_cells());
KellyErrorEstimator<dim>::estimate(
dof_handler,
solution,
estimated_error_per_cell);
- // Estimating the smoothness is performed with the method of decaing
- // expansion coefficients as outlined in the introduction.
- Vector<float> smoothness_indicators;
- SmoothnessEstimator::fourier_coefficient_decay(dof_handler,
- solution,
- smoothness_indicators);
+
+ Vector<float> smoothness_indicators(triangulation.n_active_cells());
+ estimate_smoothness(smoothness_indicators);
// Next we want to generate graphical output. In addition to the two
// estimated quantities derived above, we would also like to output the
postprocess(cycle);
}
}
+
+
+ // @sect4{LaplaceProblem::estimate_smoothness}
+
+ // As described in the introduction, we will need to take the maximum
+ // absolute value of fourier coefficients which correspond to $k$-vector
+ // $|{\bf k}|= const$. To filter the coefficients Table we
+ // will use the FESeries::process_coefficients() which requires a predicate
+ // to be specified. The predicate should operate on TableIndices and return
+ // a pair of <code>bool</code> and <code>unsigned int</code>. The latter
+ // is the value of the map from TableIndicies to unsigned int. It is
+ // used to define subsets of coefficients from which we search for the one
+ // with highest absolute value, i.e. $l^\infty$-norm. The <code>bool</code>
+ // parameter defines which indices should be used in processing. In the
+ // current case we are interested in coefficients which correspond to
+ // $0 < i*i+j*j < N*N$ and $0 < i*i+j*j+k*k < N*N$ in 2D and 3D, respectively.
+ template <int dim>
+ std::pair<bool, unsigned int>
+ LaplaceProblem<dim>::predicate(const TableIndices<dim> &ind)
+ {
+ unsigned int v = 0;
+ for (unsigned int i = 0; i < dim; i++)
+ v += ind[i] * ind[i];
+ if (v > 0 && v < max_degree * max_degree)
+ return std::make_pair(true, v);
+ else
+ return std::make_pair(false, v);
+ }
+
+ // This last function of significance implements the algorithm to estimate
+ // the smoothness exponent using the algorithms explained in detail in the
+ // introduction. We will therefore only comment on those points that are of
+ // implementational importance.
+ template <int dim>
+ void
+ LaplaceProblem<dim>::estimate_smoothness(Vector<float> &smoothness_indicators)
+ {
+ // Since most of the hard work is done for us in FESeries::Fourier and
+ // we set up the object of this class in the constructor, what we are left
+ // to do here is apply this class to calculate coefficients and then
+ // perform linear regression to fit their decay slope.
+
+
+ // First thing to do is to loop over all cells and do our work there, i.e.
+ // to locally do the Fourier transform and estimate the decay coefficient.
+ // We will use the following array as a scratch array in the loop to store
+ // local DoF values:
+ Vector<double> local_dof_values;
+
+ // Then here is the loop:
+ for (const auto &cell : dof_handler.active_cell_iterators())
+ {
+ // Inside the loop, we first need to get the values of the local
+ // degrees of freedom (which we put into the
+ // <code>local_dof_values</code> array after setting it to the right
+ // size) and then need to compute the Fourier transform by multiplying
+ // this vector with the matrix ${\cal F}$ corresponding to this finite
+ // element. This is done by calling FESeries::Fourier::calculate(),
+ // that has to be provided with the <code>local_dof_values</code>,
+ // <code>cell->active_fe_index()</code> and a Table to store
+ // coefficients.
+ local_dof_values.reinit(cell->get_fe().dofs_per_cell);
+ cell->get_dof_values(solution, local_dof_values);
+
+ fourier.calculate(local_dof_values,
+ cell->active_fe_index(),
+ fourier_coefficients);
+
+ // The next thing, as explained in the introduction, is that we wanted
+ // to only fit our exponential decay of Fourier coefficients to the
+ // largest coefficients for each possible value of $|{\bf k}|$. To
+ // this end, we use FESeries::process_coefficients() to rework
+ // coefficients into the desired format. We'll only take those Fourier
+ // coefficients with the largest magnitude for a given value of $|{\bf
+ // k}|$ and thereby need to use VectorTools::Linfty_norm:
+ std::pair<std::vector<unsigned int>, std::vector<double>> res =
+ FESeries::process_coefficients<dim>(
+ fourier_coefficients,
+ [this](const TableIndices<dim> &indices) {
+ return this->predicate(indices);
+ },
+ VectorTools::Linfty_norm);
+
+ Assert(res.first.size() == res.second.size(), ExcInternalError());
+
+ // The first vector in the <code>std::pair</code> will store values of
+ // the predicate, that is $i*i+j*j= const$ or $i*i+j*j+k*k = const$ in
+ // 2D or 3D respectively. This vector will be the same for all the cells
+ // so we can calculate logarithms of the corresponding Fourier vectors
+ // $|{\bf k}|$ only once in the whole hp-refinement cycle:
+ if (ln_k.size() == 0)
+ {
+ ln_k.resize(res.first.size(), 0);
+ for (unsigned int f = 0; f < ln_k.size(); f++)
+ ln_k[f] =
+ std::log(2.0 * numbers::PI * std::sqrt(1. * res.first[f]));
+ }
+
+ // We have to calculate the logarithms of absolute values of
+ // coefficients and use it in a linear regression fit to obtain $\mu$.
+ for (double &residual_element : res.second)
+ residual_element = std::log(residual_element);
+
+ std::pair<double, double> fit =
+ FESeries::linear_regression(ln_k, res.second);
+
+ // The final step is to compute the Sobolev index $s=\mu-\frac d2$ and
+ // store it in the vector of estimated values for each cell:
+ smoothness_indicators(cell->active_cell_index()) =
+ -fit.first - 1. * dim / 2;
+ }
+ }
} // namespace Step27
//
// ---------------------------------------------------------------------
-#ifndef dealii_fe_series_H
-#define dealii_fe_series_H
+#ifndef dealii_fe_series_h
+#define dealii_fe_series_h
using CoefficientType = typename std::complex<double>;
/**
- * A non-default constructor. The @p size_in_each_direction defines the number
- * of modes in each direction, @p fe_collection is the hp::FECollection
+ * Default constructor.
+ *
+ * Leaves all members in an empty or uninitialized state. Please call
+ * initialize() once you have all data structures ready.
+ */
+ Fourier() = default;
+
+ /**
+ * Constructor that initializes all required data structures.
+ *
+ * The @p n_coefficients_per_direction defines the number of coefficients in
+ * each direction, @p fe_collection is the hp::FECollection for which
+ * expansion will be used and @p q_collection is the hp::QCollection used to
+ * integrate the expansion for each FiniteElement in @p fe_collection.
+ */
+ Fourier(const std::vector<unsigned int> & n_coefficients_per_direction,
+ const hp::FECollection<dim, spacedim> &fe_collection,
+ const hp::QCollection<dim> & q_collection);
+
+ /**
+ * A non-default constructor. The @p n_coefficients_per_direction defines the
+ * number of modes in each direction, @p fe_collection is the hp::FECollection
* for which expansion will be used and @p q_collection is the hp::QCollection
* used to integrate the expansion for each FiniteElement
* in @p fe_collection.
+ *
+ * @deprecated Use a different constructor instead.
*/
- Fourier(const unsigned int size_in_each_direction,
+ DEAL_II_DEPRECATED
+ Fourier(const unsigned int n_coefficients_per_direction,
const hp::FECollection<dim, spacedim> &fe_collection,
const hp::QCollection<dim> & q_collection);
+ /**
+ * Initialize and overwrite all mandatory data structures for the
+ * calculation calculation of transformation matrices.
+ *
+ * All previously calculated transformation matrices will be cleared.
+ */
+ void
+ initialize(const std::vector<unsigned int> &n_coefficients_per_direction,
+ const hp::FECollection<dim, spacedim> &fe_collection,
+ const hp::QCollection<dim> & q_collection);
+
/**
* Calculate @p fourier_coefficients of the cell vector field given by
* @p local_dof_values corresponding to FiniteElement with
const unsigned int cell_active_fe_index,
Table<dim, CoefficientType> & fourier_coefficients);
+ /**
+ * Return the number of coefficients in each coordinate direction for the
+ * finite element associated with @p index in the provided hp::FECollection.
+ */
+ unsigned int
+ get_n_coefficients_per_direction(const unsigned int index) const;
+
/**
* Calculate all transformation matrices to transfer the finite element
* solution to the series expansion representation.
operator==(const Fourier<dim, spacedim> &fourier) const;
private:
+ /**
+ * Number of coefficients in each direction for each finite element in the
+ * registered hp::FECollection.
+ */
+ std::vector<unsigned int> n_coefficients_per_direction;
+
/**
* hp::FECollection for which transformation matrices will be calculated.
*/
std::vector<CoefficientType> unrolled_coefficients;
};
+
+
/**
* A class to calculate expansion of a scalar FE field into series of Legendre
* functions on a reference element.
public:
using CoefficientType = double;
+ /**
+ * Default constructor.
+ *
+ * Leaves all members in an empty or uninitialized state. Please call
+ * initialize() once you have all data structures ready.
+ */
+ Legendre() = default;
+
+ /**
+ * Constructor that initializes all required data structures.
+ *
+ * The @p n_coefficients_per_direction defines the number of coefficients in
+ * each direction, @p fe_collection is the hp::FECollection for which
+ * expansion will be used and @p q_collection is the hp::QCollection used to
+ * integrate the expansion for each FiniteElement in @p fe_collection.
+ */
+ Legendre(const std::vector<unsigned int> &n_coefficients_per_direction,
+ const hp::FECollection<dim, spacedim> &fe_collection,
+ const hp::QCollection<dim> & q_collection);
+
/**
* A non-default constructor. The @p size_in_each_direction defines the number
* of coefficients in each direction, @p fe_collection is the hp::FECollection
* for which expansion will be used and @p q_collection is the hp::QCollection
- * used to integrate the expansion for each FiniteElement
- * in @p fe_collection.
+ * used to integrate the expansion for each FiniteElement in @p fe_collection.
+ *
+ * @deprecated Use a different constructor instead.
*/
- Legendre(const unsigned int size_in_each_direction,
+ DEAL_II_DEPRECATED
+ Legendre(const unsigned int n_coefficients_per_direction,
const hp::FECollection<dim, spacedim> &fe_collection,
const hp::QCollection<dim> & q_collection);
+ /**
+ * Initialize and overwrite all mandatory data structures for the
+ * calculation of transformation matrices.
+ *
+ * All previously calculated transformation matrices will be cleared.
+ */
+ void
+ initialize(const std::vector<unsigned int> &n_coefficients_per_direction,
+ const hp::FECollection<dim, spacedim> &fe_collection,
+ const hp::QCollection<dim> & q_collection);
+
/**
* Calculate @p legendre_coefficients of the cell vector field given by
* @p local_dof_values corresponding to FiniteElement with
Table<dim, CoefficientType> & legendre_coefficients);
/**
- * Return number of coefficients in each coordinate direction.
+ * Return the number of coefficients in each coordinate direction for the
+ * finite element associated with @p index in the provided hp::FECollection.
*/
unsigned int
- get_size_in_each_direction() const;
+ get_n_coefficients_per_direction(const unsigned int index) const;
/**
* Calculate all transformation matrices to transfer the finite element
private:
/**
- * Number of coefficients in each direction
+ * Number of coefficients in each direction for each finite element in the
+ * registered hp::FECollection.
*/
- const unsigned int N;
+ std::vector<unsigned int> n_coefficients_per_direction;
/**
* hp::FECollection for which transformation matrices will be calculated.
};
+
/**
* Calculate the @p norm of subsets of @p coefficients defined by
* @p predicate being constant. Return the pair of vectors of predicate values
* used in calculation, whereas the latter is the unrolled value of indices
* according to which the subsets of coefficients will be formed.
*
- * @note Only the following values of @p norm are implemented and make sense
- * in this case: mean, L1_norm, L2_norm, Linfty_norm. The mean norm can only
- * be applied to real valued coefficients.
+ * Only those coefficients will be considered which are larger than
+ * @p smallest_abs_coefficient.
+ *
+ * @note Only the following values of @p norm_type are implemented and make
+ * sense in this case: mean, L1_norm, L2_norm, Linfty_norm. The mean norm ca
+ * only be applied to real valued coefficients.
*/
template <int dim, typename CoefficientType>
std::pair<std::vector<unsigned int>, std::vector<double>>
process_coefficients(const Table<dim, CoefficientType> &coefficients,
const std::function<std::pair<bool, unsigned int>(
const TableIndices<dim> &)> & predicate,
- const VectorTools::NormType norm);
-
-
+ const VectorTools::NormType norm_type,
+ const double smallest_abs_coefficient = 1e-10);
/**
* Linear regression least-square fit of $y = k \, x + b$.
/*@}*/
+
+
#ifndef DOXYGEN
// ------------------- inline and template functions ----------------
pred_to_values[pred_value].push_back(coeff_value);
}
+
+
template <typename CoefficientType>
void
fill_map(
}
}
+
+
template <typename CoefficientType>
void
fill_map(
}
}
+
+
template <typename CoefficientType>
void
fill_map(
}
+
template <typename Number>
double
complex_mean_value(const Number &value)
return value;
}
+
+
template <typename Number>
double
complex_mean_value(const std::complex<Number> &value)
} // namespace internal
+
template <int dim, typename CoefficientType>
std::pair<std::vector<unsigned int>, std::vector<double>>
FESeries::process_coefficients(
const Table<dim, CoefficientType> &coefficients,
const std::function<std::pair<bool, unsigned int>(const TableIndices<dim> &)>
& predicate,
- const VectorTools::NormType norm)
+ const VectorTools::NormType norm_type,
+ const double smallest_abs_coefficient)
{
+ Assert(smallest_abs_coefficient >= 0.,
+ ExcMessage("smallest_abs_coefficient should be non-negative."));
+
std::vector<unsigned int> predicate_values;
std::vector<double> norm_values;
// now go through the map and populate the @p norm_values based on @p norm:
for (const auto &pred_to_value : pred_to_values)
{
- predicate_values.push_back(pred_to_value.first);
Vector<CoefficientType> values(pred_to_value.second.cbegin(),
pred_to_value.second.cend());
- switch (norm)
+ double norm_value = 0;
+ switch (norm_type)
{
case VectorTools::L2_norm:
{
- norm_values.push_back(values.l2_norm());
+ norm_value = values.l2_norm();
break;
}
case VectorTools::L1_norm:
{
- norm_values.push_back(values.l1_norm());
+ norm_value = values.l1_norm();
break;
}
case VectorTools::Linfty_norm:
{
- norm_values.push_back(values.linfty_norm());
+ norm_value = values.linfty_norm();
break;
}
case VectorTools::mean:
{
- norm_values.push_back(
- internal::FESeriesImplementation::complex_mean_value(
- values.mean_value()));
+ norm_value = internal::FESeriesImplementation::complex_mean_value(
+ values.mean_value());
break;
}
default:
AssertThrow(false, ExcNotImplemented());
break;
}
+
+ // will use all non-zero coefficients
+ if (std::abs(norm_value) > smallest_abs_coefficient)
+ {
+ predicate_values.push_back(pred_to_value.first);
+ norm_values.push_back(norm_value);
+ }
}
return std::make_pair(predicate_values, norm_values);
// Store information about those resources which have been used to generate
// the transformation matrices.
// mode vector
- unsigned int size = k_vectors.n_elements();
- ar & size;
+ ar &n_coefficients_per_direction;
// finite element collection
- size = fe_collection->size();
- ar &size;
+ unsigned int size = fe_collection->size();
+ ar & size;
for (unsigned int i = 0; i < size; ++i)
ar &(*fe_collection)[i].get_name();
// Check whether the currently registered resources are compatible with
// the transformation matrices to load.
// mode vector
- unsigned int size;
- ar & size;
- AssertDimension(size, k_vectors.n_elements());
+ std::vector<unsigned int> compare_coefficients;
+ ar & compare_coefficients;
+ Assert(compare_coefficients == n_coefficients_per_direction,
+ ExcMessage("A different number of coefficients vector has been used "
+ "to generate the transformation matrices you are about "
+ "to load!"));
// finite element collection
- ar &size;
+ unsigned int size;
+ ar & size;
AssertDimension(size, fe_collection->size());
std::string name;
for (unsigned int i = 0; i < size; ++i)
// Store information about those resources which have been used to generate
// the transformation matrices.
// mode vector
- unsigned int size = N;
- ar & size;
+ ar &n_coefficients_per_direction;
// finite element collection
- size = fe_collection->size();
- ar &size;
+ unsigned int size = fe_collection->size();
+ ar & size;
for (unsigned int i = 0; i < size; ++i)
ar &(*fe_collection)[i].get_name();
// Check whether the currently registered resources are compatible with
// the transformation matrices to load.
// mode vector
- unsigned int size;
- ar & size;
- AssertDimension(size, N);
+ std::vector<unsigned int> compare_coefficients;
+ ar & compare_coefficients;
+ Assert(compare_coefficients == n_coefficients_per_direction,
+ ExcMessage("A different number of coefficients vector has been used "
+ "to generate the transformation matrices you are about "
+ "to load!"));
// finite element collection
- ar &size;
+ unsigned int size;
+ ar & size;
AssertDimension(size, fe_collection->size());
std::string name;
for (unsigned int i = 0; i < size; ++i)
DEAL_II_NAMESPACE_CLOSE
-#endif // dealii_fe_series_H
+#endif // dealii_fe_series_h
n_blocks() const;
/**
- * Return the maximal degree over all elements of this collection.
+ * Return the maximum of values returned by FiniteElement::get_degree()
+ * over all elements of this collection.
*/
unsigned int
max_degree() const;
// ---------------------------------------------------------------------
//
-// Copyright (C) 1998 - 2018 by the deal.II authors
+// Copyright (C) 2018 - 2020 by the deal.II authors
//
// This file is part of the deal.II library.
//
#include <deal.II/base/config.h>
-#include <deal.II/dofs/dof_accessor.h>
-#include <deal.II/dofs/dof_handler.h>
-
-#include <deal.II/fe/fe_series.h>
-
-#include <deal.II/hp/dof_handler.h>
-
-#include <deal.II/lac/vector.h>
+#include <deal.II/fe/component_mask.h>
#include <deal.II/numerics/vector_tools.h>
+#include <functional>
#include <vector>
DEAL_II_NAMESPACE_OPEN
+// forward declarations
+#ifndef DOXYGEN
+template <typename Number>
+class Vector;
+
+namespace FESeries
+{
+ template <int dim, int spacedim>
+ class Fourier;
+ template <int dim, int spacedim>
+ class Legendre;
+} // namespace FESeries
+
+namespace hp
+{
+ template <int dim, int spacedim>
+ class DoFHandler;
+ template <int dim, int spacedim>
+ class FECollection;
+} // namespace hp
+#endif
+
+
/**
* A namespace for various smoothness estimation strategies for hp-adaptive FEM.
+ *
+ * Smoothness estimation is one strategy to decide whether a cell with a large
+ * error estimate should undergo h- or p-refinement. Typical strategies decide
+ * to increase the polynomial degree on a cell if the solution is particularly
+ * smooth, whereas one would refine the mesh if the solution on the cell is
+ * singular, has kinks in some derivative, or is otherwise not particularly
+ * smooth. All of these strategies rely on a way to identify how "smooth" a
+ * function is on a given cell.
*/
namespace SmoothnessEstimator
{
/**
- * Estimate smoothness from decay of Legendre absolute values of coefficients
- * on the reference cell.
+ * Smoothness estimation strategy based on the decay of Legendre expansion
+ * coefficients.
*
- * In one dimension, the finite element solution on the reference element with
- * polynomial degree $p$ can be written as
+ * In one dimension, the finite element solution on cell $K$ with polynomial
+ * degree $p$ can be written as
+ * @f{eqnarray*}
+ * u_h(x) &=& \sum_j u_j \varphi_j (x) \\
+ * u_{h, k}(x) &=& \sum_{k=0}^{p} a_k \widetilde P_k (x),
+ * \quad a_k = \sum_j {\cal L}_{k,j} u_j
+ * @f}
+ * where $u_j$ are degrees of freedom and $\varphi_j$ are the corresponding
+ * shape functions. $\{\widetilde P_k(x)\}$ are Legendre polynomials on cell
+ * $K$. $a_k$ and ${\cal L}_{k,j}$ are coefficients and transformation
+ * matrices from the Legendre expansion of each shape function. For practical
+ * reasons, we will perform the calculation of these matrices and coefficients
+ * only on the reference cell $\hat K$. We only have to calculate the
+ * transformation matrices once this way. However, results are only applicable
+ * if mapping from the reference cell to the actual cell is linear. We use
+ * the class FESeries::Legendre to determine all coefficients $a_k$.
+ *
+ * A function is analytic, i.e., representable by a power series, if and only
+ * if their Legendre expansion coefficients decay as (see @cite eibner2007hp)
* @f[
- * u_h(\hat x) = \sum_{j=0}^{p} a_j P_j(\hat x)
+ * |a_k| \sim c \, \exp(-\sigma k)
* @f]
- * where $\{P_j(x)\}$ are Legendre polynomials. The decay of the coefficients
- * is estimated by performing the linear regression fit of
+ * We determine their decay rate $\sigma$ by performing the linear regression
+ * fit of
* @f[
- * \ln |a_j| \sim C - \sigma j
+ * \ln |a_k| \sim C - \sigma k
* @f]
- * for $j=0,..,p$. The rate of the decay $\sigma$ can be used to estimate the
- * smoothness. For example, one strategy to implement hp-refinement
- * criteria is to perform p-refinement if $\sigma>1$.
- *
- * Extension to higher dimension is done by performing the fit in each
- * coordinate direction separately and then taking the lowest value of
- * $\sigma$.
- *
- * For each input vector of degrees of freedom defined on a DoFHandler,
- * this function returns a vector with as many elements as there are cells
- * where each element contains $\exp(-\sigma)$, which is a so-called
- * analyticity (see references below).
- *
- * @param [in] fe_series FESeries::Legendre object to calculate coefficients.
- * This object needs to be initialized to have at least $p+1$ coefficients in
- * each direction, where $p$ is the maximum polynomial degree to be used.
- * @param [in] dof_hander An hp::DoFHandler
- * @param [in] all_solutions A vector of pointers to the solution vectors
- * @param [out] all_smoothness_indicators A vector of pointers to the smoothness indicators for each @p all_solutions.
- * @param [in] coefficients_predicate A predicate to select Legendre
- * coefficients $a_j \;\; j=0\dots p$ for linear regression in each coordinate
- * direction. The user is responsible for updating the vector of `flags`
- * provided to this function. Note that its size is $p+1$, where $p$ is the
- * polynomial degree of the FE basis on a given element. Default
- * implementation will use all Legendre coefficients in each coordinate
- * direction, i.e. set all elements of the vector to `true`.
- * @param [in] smallest_abs_coefficient The smallest absolute value of the
- * coefficient to be used in linear regression in each coordinate direction.
- * Note that Legendre coefficients of some functions may have a repeating
- * pattern of zero coefficients (i.e. for functions that are locally symmetric
- * or antisymmetric about the midpoint of the element in any coordinate
- * direction). Thus this parameters allows to ingore small (in absolute value)
- * coefficients within the linear regression fit. In case there are less than
- * two non-zero coefficients for a coordinate direction, this direction will
- * be skipped. If all coefficients are zero, the returned value for this cell
- * will be zero (i.e. corresponding to the $\sigma=\infty$).
- *
- * For more theoretical details see
- * @code{.bib}
- * @Article{Mavriplis1994,
- * author = {Mavriplis, Catherine},
- * title = {Adaptive mesh strategies for the spectral element method},
- * journal = {{Computer Methods in Applied Mechanics and Engineering}},
- * year = {1994},
- * volume = {116},
- * number = {1},
- * pages = {77--86},
- * publisher = {Elsevier},
- * }
- * @article{Houston2005,
- * author = {Houston, Paul and S{\"u}li, Endre},
- * title = {A note on the design of hp-adaptive finite element
- * methods for elliptic partial differential equations},
- * journal = {{Computer Methods in Applied Mechanics and Engineering}},
- * number = {2},
- * pages = {229--243},
- * publisher = {Elsevier},
- * volume = {194},
- * year = {2005}
- * }
- * @article{Eibner2007,
- * author = {Eibner, Tino and Melenk, Jens Markus},
- * title = {An adaptive strategy for hp-FEM based on testing for
- * analyticity},
- * journal = {{Computational Mechanics}},
- * year = {2007},
- * volume = {39},
- * number = {5},
- * pages = {575--595},
- * publisher = {Springer},
- * }
- * @endcode
- * and for the application within the deal.II:
- * @code{.bib}
- * @article{Davydov2017,
- * author = {Denis Davydov and Tymofiy Gerasimov and Jean-Paul Pelteret and
- * Paul Steinmann},
- * title = {Convergence study of the h-adaptive PUM and the hp-adaptive
- * FEM applied to eigenvalue problems in quantum mechanics},
- * journal = {{Advanced Modeling and Simulation in Engineering Sciences}},
- * year = {2017},
- * volume = {4},
- * number = {1},
- * pages = {7},
- * issn = {2213-7467},
- * doi = {10.1186/s40323-017-0093-0},
- * }
- * @endcode
+ * for $k=0,\ldots,p$, with $p$ the polynomial degree of the finite element.
+ * The rate of the decay $\sigma$ can be used to estimate the smoothness. For
+ * example, one strategy to implement hp-refinement criteria is to perform
+ * p-refinement if $\sigma>1$ (see @cite mavriplis1994hp).
*
- * @ingroup numerics
- * @author Denis Davydov, 2018
+ * @author Denis Davydov, 2018, Marc Fehling, 2018 - 2020
*/
- template <int dim, int spacedim, typename VectorType>
- void
- legendre_coefficient_decay(
- FESeries::Legendre<dim, spacedim> & fe_series,
- const hp::DoFHandler<dim, spacedim> & dof_handler,
- const std::vector<const VectorType *> &all_solutions,
- const std::vector<Vector<float> *> & all_smoothness_indicators,
- const std::function<void(std::vector<bool> &flags)> coefficients_predicate =
- [](std::vector<bool> &flags) -> void {
- std::fill(flags.begin(), flags.end(), true);
- },
- const double smallest_abs_coefficient = 1e-10);
+ namespace Legendre
+ {
+ /**
+ * In this variant of the estimation strategy for higher dimensions, we will
+ * consider all mode vectors $\bf k$ describing Legendre polynomials
+ * $\widetilde P_{\bf k}$ and perform one least-squares fit over all
+ * coefficients at once. If there are multiple coefficients corresponding to
+ * the same absolute value of modes $\|{\bf k}\|_1$, we take the maximum
+ * among those. Thus, the least-squares fit is performed on
+ * @f{eqnarray*}
+ * \widetilde P_{\bf k}({\bf x}) &=&
+ * \widetilde P_{k_1} (x_1) \ldots \widetilde P_{k_d} (x_d) \\
+ * \ln \left( \max\limits_{\|{\bf k}\|_1} |a_{\bf k}| \right) &\sim&
+ * C - \sigma \|{\bf k}\|_1
+ * @f}
+ * for ${\bf k}=(k_1,\ldots,k_d)$ and $k_i=0,\ldots,p$, with $p$ the
+ * polynomial degree of the finite element.
+ *
+ * For a finite element approximation @p solution, this function writes the
+ * decay rate for every cell into the output vector @p smoothness_indicators.
+ *
+ * @param [in] fe_legendre FESeries::Legendre object to calculate coefficients.
+ * This object needs to be initialized to have at least $p+1$ coefficients
+ * in each direction for every finite element in the collection, where $p$
+ * is its polynomial degree.
+ * @param [in] dof_handler An hp::DoFHandler.
+ * @param [in] solution A solution vector.
+ * @param [out] smoothness_indicators A vector for smoothness indicators.
+ * @param [in] regression_strategy Determines which norm will be used on the
+ * subset of coefficients $\mathbf{k}$ with the same absolute value
+ * $\|{\bf k}\|_1$. Default is VectorTools::Linfty_norm for a maximum
+ * approximation.
+ * @param [in] smallest_abs_coefficient The smallest absolute value of the
+ * coefficient to be used in linear regression. Note that Legendre
+ * coefficients of some functions may have a repeating pattern of zero
+ * coefficients (i.e. for functions that are locally symmetric or
+ * antisymmetric about the midpoint of the element in any coordinate
+ * direction). Thus this parameters allows to ingore small (in absolute
+ * value) coefficients within the linear regression fit. In case there are
+ * less than two nonzero coefficients, the returned value for this cell will
+ * be $\sigma=\infty$.
+ * @param [in] only_flagged_cells Smoothness indicators are usually used to
+ * decide whether to perform h- or p-adaptation. So in most cases, we only
+ * need to calculate those indicators on cells flagged for refinement or
+ * coarsening. This parameter controls whether this particular subset or all
+ * cells will be considered. By default, all cells will be considered. When
+ * only flagged cells are supposed to be considered, smoothness indicators
+ * will only be set on those vector entries of flagged cells; the others
+ * will be set to a signaling NaN.
+ *
+ * For more theoretical details see @cite mavriplis1994hp
+ * @cite houston2005hp @cite eibner2007hp.
+ */
+ template <int dim, int spacedim, typename VectorType>
+ void
+ coefficient_decay(FESeries::Legendre<dim, spacedim> & fe_legendre,
+ const hp::DoFHandler<dim, spacedim> &dof_handler,
+ const VectorType & solution,
+ Vector<float> & smoothness_indicators,
+ const VectorTools::NormType regression_strategy =
+ VectorTools::Linfty_norm,
+ const double smallest_abs_coefficient = 1e-10,
+ const bool only_flagged_cells = false);
+
+ /**
+ * In this variant of the estimation strategy for higher dimensions, we only
+ * consider modes in each coordinate direction, i.e., only mode vectors
+ * $\bf k$ with one nonzero entry. We perform the least-squares fit in
+ * each coordinate direction separately and take the lowest decay rate
+ * $\sigma$ among them.
+ *
+ * For a finite element approximation @p solution, this function writes the
+ * decay rate for every cell into the output vector @p smoothness_indicators.
+ *
+ * @param [in] fe_legendre FESeries::Legendre object to calculate coefficients.
+ * This object needs to be initialized to have at least $p+1$ coefficients
+ * in each direction, where $p$ is the maximum polynomial degree to be used.
+ * @param [in] dof_handler An hp::DoFHandler
+ * @param [in] solution A solution vector
+ * @param [out] smoothness_indicators A vector for smoothness indicators
+ * @param [in] coefficients_predicate A predicate to select Legendre
+ * coefficients $a_j$, $j=0,\ldots,p$ for linear regression in each
+ * coordinate direction. The user is responsible for updating the vector of
+ * `flags` provided to this function. Note that its size is $p+1$, where $p$
+ * is the polynomial degree of the FE basis on a given element. The default
+ * implementation will use all Legendre coefficients in each coordinate
+ * direction, i.e. set all elements of the vector to `true`.
+ * @param [in] smallest_abs_coefficient The smallest absolute value of the
+ * coefficient to be used in linear regression in each coordinate direction.
+ * Note that Legendre coefficients of some functions may have a repeating
+ * pattern of zero coefficients (i.e. for functions that are locally
+ * symmetric or antisymmetric about the midpoint of the element in any
+ * coordinate direction). Thus this parameters allows to ingore small (in
+ * absolute value) coefficients within the linear regression fit. In case
+ * there are less than two nonzero coefficients for a coordinate direction,
+ * this direction will be skipped. If all coefficients are zero, the
+ * returned value for this cell will be $\sigma=\infty$.
+ * @param [in] only_flagged_cells Smoothness indicators are usually used to
+ * decide whether to perform h- or p-adaptation. So in most cases, we only
+ * need to calculate those indicators on cells flagged for refinement or
+ * coarsening. This parameter controls whether this particular subset or all
+ * cells will be considered. By default, all cells will be considered. When
+ * only flagged cells are supposed to be considered, smoothness indicators
+ * will only be set on those vector entries of flagged cells; the others
+ * will be set to NaN.
+ *
+ * For more theoretical details and the application within the deal.II
+ * library see @cite davydov2017hp.
+ */
+ template <int dim, int spacedim, typename VectorType>
+ void
+ coefficient_decay_per_direction(
+ FESeries::Legendre<dim, spacedim> & fe_legendre,
+ const hp::DoFHandler<dim, spacedim> &dof_handler,
+ const VectorType & solution,
+ Vector<float> & smoothness_indicators,
+ const ComponentMask &coefficients_predicate = ComponentMask(),
+ const double smallest_abs_coefficient = 1e-10,
+ const bool only_flagged_cells = false);
+
+ /**
+ * Number of modes for the default configuration of the Legendre series
+ * expansion FESeries::Legendre.
+ *
+ * We use as many modes as the highest polynomial of all finite elements
+ * used plus one, since we start with the first Legendre polynomial which is
+ * just a constant.
+ */
+ template <int dim, int spacedim>
+ std::vector<unsigned int>
+ default_number_of_coefficients_per_direction(
+ const hp::FECollection<dim, spacedim> &fe_collection);
+
+ /**
+ * Quadrature collection for the default configuration of the Legendre
+ * series expansion FESeries::Legendre.
+ *
+ * We use Gaussian quadrature designed to yield exact results for the
+ * highest order Legendre polynomial used.
+ */
+ template <int dim, int spacedim>
+ hp::QCollection<dim>
+ default_quadrature_collection(
+ const hp::FECollection<dim, spacedim> &fe_collection);
+ } // namespace Legendre
- /**
- * Same as above, but for a single solution vector.
- */
- template <int dim, int spacedim, typename VectorType>
- void
- legendre_coefficient_decay(
- FESeries::Legendre<dim, spacedim> & fe_series,
- const hp::DoFHandler<dim, spacedim> & dof_handler,
- const VectorType & solution,
- Vector<float> & smoothness_indicators,
- const std::function<void(std::vector<bool> &flags)> coefficients_predicate =
- [](std::vector<bool> &flags) -> void {
- std::fill(flags.begin(), flags.end(), true);
- },
- const double smallest_abs_coefficient = 1e-10);
- /**
- * Same as above, but for a single solution vector and with the default
- * FESeries::Legendre.
- */
- template <int dim, int spacedim, typename VectorType>
- void
- legendre_coefficient_decay(
- const hp::DoFHandler<dim, spacedim> & dof_handler,
- const VectorType & solution,
- Vector<float> & smoothness_indicators,
- const std::function<void(std::vector<bool> &flags)> coefficients_predicate =
- [](std::vector<bool> &flags) -> void {
- std::fill(flags.begin(), flags.end(), true);
- },
- const double smallest_abs_coefficient = 1e-10);
/**
- * Estimate the smoothness of a solution based on the decay of coefficients
- * from a series expansion.
+ * Smoothness estimation strategy based on the decay of Fourier expansion
+ * coefficients.
*
- * From the definition, we can write our series expansion $\hat U_{\bf k}$ as
- * a matrix product
+ * From the definition, we can write our Fourier series expansion
+ * $a_{\bf k}$ of the finite element solution on cell $K$ with polynomial
+ * degree $p$ as a matrix product
+ * @f{eqnarray*}
+ * u_h({\bf x}) &=& \sum_j u_j \varphi_j ({\bf x}) \\
+ * u_{h, {\bf k}}({\bf x}) &=&
+ * \sum_{{\bf k}, \|{\bf k}\|\le p} a_{\bf k} \phi_{\bf k}({\bf x}),
+ * \quad a_{\bf k} = \sum_j {\cal F}_{{\bf k},j} u_j
+ * @f}
+ * with $u_j$ the degrees of freedom and $\varphi_j$ the corresponding shape
+ * functions. $\{\phi_{\bf k}({\bf x}) = \exp(i \, 2 \pi \, {\bf k} \cdot
+ * {\bf x}) \}$ are expoential functions on cell $K$. $a_{\bf k}$ and ${\cal
+ * F}_{{\bf k},j}$ are coefficients and transformation matrices from the
+ * Fourier expansion of each shape function. For practical reasons, we will
+ * perform the calculation of these matrices and coefficients only on the
+ * reference cell $\hat K$. We only have to calculate the transformation
+ * matrices once this way. However, results are only applicable if mapping
+ * from the reference cell to the actual cell is linear. We use the class
+ * FESeries::Fourier to determine all coefficients $a_{\bf k}$.
+ *
+ * If the finite element approximation on cell $K$ is part of the Hilbert
+ * space $H^s(K)$, then the following integral must exit for both finite
+ * element and spectral representation of our solution
+ * @f{eqnarray*}
+ * \| \nabla^s u_h({\bf x}) \|_{L^2(K)}^2 &=&
+ * \int\limits_K \left| \nabla^s u_h({\bf x}) \right|^2 d{\bf x} <
+ * \infty \\
+ * \| \nabla^s u_{h, {\bf k}}({\bf x}) \|_{L^2(K)}^2 &=&
+ * \int\limits_K \left| \sum\limits_{\bf k} (-i \, 2 \pi {\bf k})^s \,
+ * a_{\bf k} \, \phi_{\bf k}({\bf x}) \right|^2 d{\bf x} =
+ * (2 \pi)^{2s} \sum\limits_{\bf k} \left| a_{\bf k} \right|^2
+ * \|{\bf k}\|_2^{2s} < \infty
+ * @f}
+ * The sum is finite only if the summands decay at least with order
* @f[
- * \hat U_{\bf k}
- * = {\cal F}_{{\bf k},j} u_j,
+ * |a_{\bf k}|^2 \|{\bf k}\|_2^{2s} \|{\bf k}\|_2^{d - 1} =
+ * {\cal O}\left( \|{\bf k}\|_2^{-1-\epsilon} \right)
+ * @f]
+ * for all $\epsilon > 0$. The additional factor stems from the fact that,
+ * since we sum over all multi-indices ${\bf k}$ that are located on a
+ * dim-dimensional sphere, we actually have, up to a constant,
+ * $\|{\bf k}\|_2^{d-1}$ modes located in each increment $\|{\bf k}\|_2 +
+ * d\|{\bf k}\|_2$ that need to be taken into account. By a comparison of
+ * exponents, we can rewrite this condition as
+ * @f[
+ * |a_{\bf k}| = {\cal O}\left(\|{\bf k}\|_2^
+ * {-\left(s + \frac d2 + \epsilon \right)} \right)
* @f]
- * with $u_j$ the coefficients and ${\cal F}_{{\bf k},j}$ the transformation
- * matrix from the expansion. We use the classes FESeries::Fourier and
- * FESeries::Legendre to determine all coefficients $u_j$.
*
* The next step is that we have to estimate how fast these coefficients
- * decay with $|{\bf k}|$. Thus, we perform a least-squares fit
+ * decay with $\|{\bf k}\|_2$. Thus, we perform a least-squares fit
* @f[
- * \min_{\alpha,\mu}
- * \frac 12 \sum_{{\bf k}, |{\bf k}|\le N}
- * \left( |\hat U_{\bf k}| - \alpha |{\bf k}|^{-\mu}\right)^2
+ * \min_{\alpha,\sigma}
+ * \frac 12 \sum_{{\bf k}, \|{\bf k}\|_2 \le p}
+ * \left( |a_{\bf k}| - \alpha \|{\bf k}\|_2^{-\sigma}\right)^2
* @f]
- * with linear regressions coefficients $\alpha$ and $\mu$. For
- * simplification, we apply a logarithm on our minimization problem
+ * with regression coefficients $\alpha$ and $\sigma$. For simplification, we
+ * apply a logarithm on our minimization problem
* @f[
- * \min_{\beta,\mu}
- * Q(\beta,\mu) =
- * \frac 12 \sum_{{\bf k}, |{\bf k}|\le N}
- * \left( \ln |\hat U_{\bf k}| - \beta + \mu \ln |{\bf k}|\right)^2,
+ * \min_{\beta,\sigma}
+ * Q(\beta,\sigma) =
+ * \frac 12 \sum_{{\bf k}, \|{\bf k}\|_2 \le p}
+ * \left( \ln |a_{\bf k}| - \beta + \sigma \ln \|{\bf k}\|_2
+ * \right)^2,
* @f]
- * where $\beta=\ln \alpha$. This is now a problem for which the
- * optimality conditions $\frac{\partial Q}{\partial\beta}=0,
- * \frac{\partial Q}{\partial\mu}=0$, are linear in $\beta,\mu$. We can
+ * where $\beta=\ln \alpha$. This is now a problem for which the optimality
+ * conditions $\frac{\partial Q}{\partial\beta}=0,
+ * \frac{\partial Q}{\partial\sigma}=0$, are linear in $\beta,\sigma$. We can
* write these conditions as follows:
* @f[
* \left(\begin{array}{cc}
- * \sum_{{\bf k}, |{\bf k}|\le N} 1 &
- * \sum_{{\bf k}, |{\bf k}|\le N} \ln |{\bf k}|
+ * \sum_{{\bf k}, \|{\bf k}\|_2 \le p} 1 &
+ * \sum_{{\bf k}, \|{\bf k}\|_2 \le p} \ln \|{\bf k}\|_2
* \\
- * \sum_{{\bf k}, |{\bf k}|\le N} \ln |{\bf k}| &
- * \sum_{{\bf k}, |{\bf k}|\le N} (\ln |{\bf k}|)^2
+ * \sum_{{\bf k}, \|{\bf k}\|_2 \le p} \ln \|{\bf k}\|_2 &
+ * \sum_{{\bf k}, \|{\bf k}\|_2 \le p} (\ln \|{\bf k}\|_2)^2
* \end{array}\right)
* \left(\begin{array}{c}
- * \beta \\ -\mu
+ * \beta \\ -\sigma
* \end{array}\right)
* =
* \left(\begin{array}{c}
- * \sum_{{\bf k}, |{\bf k}|\le N} \ln |\hat U_{{\bf k}}|
+ * \sum_{{\bf k}, \|{\bf k}\|_2\le p} \ln |a_{{\bf k}}|
* \\
- * \sum_{{\bf k}, |{\bf k}|\le N} \ln |\hat U_{{\bf k}}| \ln |{\bf k}|
- * \end{array}\right)
+ * \sum_{{\bf k}, \|{\bf k}\|_2\le p} \ln |a_{{\bf k}}| \ln \|{\bf
+ * k}\|_2 \end{array}\right)
* @f]
- * Solving for $\beta$ and $\mu$ is nothing else but a linear regression fit
- * and to do that we will use FESeries::linear_regression().
+ * Solving for $\beta$ and $\sigma$ is just a linear regression fit and to do
+ * that we will use FESeries::linear_regression().
*
* While we are not particularly interested in the actual value of
- * $\beta$, the formula above gives us a mean to calculate the value of
- * the exponent $\mu$ that we can then use to determine that
- * $\hat u(\hat{\bf x})$ is in $H^s(\hat K)$ with $s=\mu-\frac d2$. These
- * Sobolev indices $s$ will suffice as our smoothness estimators and will be
- * calculated on each cell for any provided solution.
- *
- * @note An extensive demonstration of the use of these functions is provided in step-27.
- *
- * The @p regression_strategy parameter determines which norm will be used on the subset of
- * coeffiecients $\mathbf{k}$ with the same absolute value $|\mathbf{k}|$.
- * Default is VectorTools::Linfty_norm for a maximum approximation.
+ * $\beta$, the formula above gives us a means to calculate the value of
+ * the exponent $\sigma$ that we can then use to determine that
+ * $u(\hat{\bf x})$ is in $H^s(K)$ with $s=\sigma-\frac d2$. The
+ * decay rates $\sigma$ will suffice as our smoothness indicators and
+ * will be calculated on each cell for any provided solution.
*
- * Smoothness indicators will be calculated for each solution in @p all_solutions
- * and stored in @p all_smoothness_indicators in the same order.
- *
- * An individual @p fe_series object can be supplied, which has to be constructed with the
- * same FECollection object as the @p dof_handler.
- *
- * @ingroup numerics
- * @author Denis Davydov, 2016, Marc Fehling, 2018
+ * @note An extensive demonstration of the use of these functions is
+ * provided in step-27.
*
+ * @author Wolfgang Bangerth 2006 - 2007, Denis Davydov, 2016,
+ * Marc Fehling, 2018 - 2020
*/
- template <int dim, int spacedim, typename VectorType>
- void
- fourier_coefficient_decay(
- FESeries::Fourier<dim, spacedim> & fe_series,
- const hp::DoFHandler<dim, spacedim> & dof_handler,
- const std::vector<const VectorType *> &all_solutions,
- const std::vector<Vector<float> *> & all_smoothness_indicators,
- const VectorTools::NormType regression_strategy = VectorTools::Linfty_norm);
+ namespace Fourier
+ {
+ /**
+ * In this variant of the estimation strategy for higher dimensions, we will
+ * consider all mode vectors $\bf k$ describing Fourier polynomials
+ * $P_{\bf k}$ and perform one least-squares fit over all coefficients
+ * at once. If there are multiple coefficients corresponding to the same
+ * absolute value of modes $\|\bf k\|_2$, we take the maximum among those.
+ * Thus, the least-squares fit is performed on
+ * @f[
+ * \ln \left( \max\limits_{\|{\bf k}\|_2} |a_{\bf k}| \right) \sim
+ * C - \sigma \ln \|{\bf k}\|_2
+ * @f]
+ * for ${\bf k}=(k_1,\ldots,k_d)$ and $k_i=0,\ldots,p$, with $p$ the
+ * polynomial degree of the finite element. We exclude the $\|{\bf k}\|_2=0$
+ * modes to avoid the singularity of the logarithm.
+ *
+ * The @p regression_strategy parameter determines which norm will be used
+ * on the subset of coefficients $\bf k$ with the same absolute value
+ * $\|{\bf k}\|_2$. Default is VectorTools::Linfty_norm for a maximum
+ * approximation.
+ *
+ * For a provided solution vector @p solution defined on a DoFHandler
+ * @p dof_handler, this function returns a vector @p smoothness_indicators
+ * with as many elements as there are cells where each element contains the
+ * estimated regularity $\sigma$.
+ *
+ * A series expansion object @p fe_fourier has to be supplied, which needs
+ * to be constructed with the same FECollection object as the @p dof_handler.
+ *
+ * The parameter @p smallest_abs_coefficient allows to ingore small (in
+ * absolute value) coefficients within the linear regression fit. In case
+ * there are less than two nonzero coefficients for a coordinate direction,
+ * this direction will be skipped. If all coefficients are zero, the
+ * returned value for this cell will be $\sigma=\infty$.
+ *
+ * Smoothness indicators are usually used to decide whether to perform h- or
+ * p-adaptation. So in most cases, we only need to calculate those
+ * indicators on cells flagged for refinement or coarsening. The parameter
+ * @p only_flagged_cells controls whether this particular subset or all
+ * cells will be considered. By default, all cells will be considered.
+ * When only flagged cells are supposed to be considered, smoothness
+ * indicators will only be set on those vector entries of flagged cells;
+ * the others will be set to a signaling NaN.
+ */
+ template <int dim, int spacedim, typename VectorType>
+ void
+ coefficient_decay(FESeries::Fourier<dim, spacedim> & fe_fourier,
+ const hp::DoFHandler<dim, spacedim> &dof_handler,
+ const VectorType & solution,
+ Vector<float> & smoothness_indicators,
+ const VectorTools::NormType regression_strategy =
+ VectorTools::Linfty_norm,
+ const double smallest_abs_coefficient = 1e-10,
+ const bool only_flagged_cells = false);
- /**
- * Same as the function above, only for one @p solution vector.
- */
- template <int dim, int spacedim, typename VectorType>
- void
- fourier_coefficient_decay(
- FESeries::Fourier<dim, spacedim> & fe_series,
- const hp::DoFHandler<dim, spacedim> &dof_handler,
- const VectorType & solution,
- Vector<float> & smoothness_indicators,
- const VectorTools::NormType regression_strategy = VectorTools::Linfty_norm);
+ /**
+ * In this variant of the estimation strategy for higher dimensions, we only
+ * consider modes in each coordinate direction, i.e., only mode vectors
+ * $\bf k$ with one nonzero entry. We perform the least-squares fit in
+ * each coordinate direction separately and take the lowest decay rate
+ * $\sigma$ among them.
+ *
+ * The @p coefficients_predicate parameter selects Fourier coefficients
+ * $a_j$, $j=0,\ldots,p$ for linear regression in each coordinate
+ * direction. The user is responsible for updating the vector of `flags`
+ * provided to this function. Note that its size is $p+1$, where $p$ is the
+ * polynomial degree of the FE basis on a given element. The default
+ * implementation will use all Fourier coefficients in each coordinate
+ * direction, i.e. set all elements of the vector to `true`.
+ *
+ * For a provided solution vector @p solution defined on a DoFHandler
+ * @p dof_handler, this function returns a vector @p smoothness_indicators
+ * with as many elements as there are cells where each element contains the
+ * estimated regularity $\sigma$.
+ *
+ * A series expansion object @p fe_fourier has to be supplied, which needs
+ * to be constructed with the same FECollection object as the @p dof_handler.
+ *
+ * The parameter @p smallest_abs_coefficient allows to ingore small (in
+ * absolute value) coefficients within the linear regression fit. In case
+ * there are less than two nonzero coefficients for a coordinate direction,
+ * this direction will be skipped. If all coefficients are zero, the
+ * returned value for this cell will be $\sigma=\infty$.
+ *
+ * Smoothness indicators are usually used to decide whether to perform h- or
+ * p-adaptation. So in most cases, we only need to calculate those
+ * indicators on cells flagged for refinement or coarsening. The parameter
+ * @p only_flagged_cells controls whether this particular subset or all
+ * cells will be considered. By default, all cells will be considered.
+ * When only flagged cells are supposed to be considered, smoothness
+ * indicators will only be set on those vector entries of flagged cells;
+ * the others will be set to a signaling NaN.
+ */
+ template <int dim, int spacedim, typename VectorType>
+ void
+ coefficient_decay_per_direction(
+ FESeries::Fourier<dim, spacedim> & fe_fourier,
+ const hp::DoFHandler<dim, spacedim> &dof_handler,
+ const VectorType & solution,
+ Vector<float> & smoothness_indicators,
+ const ComponentMask &coefficients_predicate = ComponentMask(),
+ const double smallest_abs_coefficient = 1e-10,
+ const bool only_flagged_cells = false);
- /**
- * Same as the function above, but with a default configuration for the chosen
- * series expansion, using 2-point Gaussian quadrature for each finite
- * element.
- *
- * Provide the desired series expansion as a template argument, i.e.
- * @code
- * SmoothnessEstimator::estimate_by_coefficient_decay<FESeries::Fourier<dim>>(
- * dof_handler, all_solutions, all_smoothness_indicators);
- * @endcode
- */
- template <int dim, int spacedim, typename VectorType>
- void
- fourier_coefficient_decay(
- const hp::DoFHandler<dim, spacedim> & dof_handler,
- const std::vector<const VectorType *> &all_solutions,
- const std::vector<Vector<float> *> & all_smoothness_indicators,
- const VectorTools::NormType regression_strategy = VectorTools::Linfty_norm);
+ /**
+ * Number of modes for the default configuration of the Fourier series
+ * expansion FESeries::Fourier.
+ *
+ * We use as many modes as the highest polynomial degree of all finite
+ * elements used plus one, and at least three modes.
+ */
+ template <int dim, int spacedim>
+ std::vector<unsigned int>
+ default_number_of_coefficients_per_direction(
+ const hp::FECollection<dim, spacedim> &fe_collection);
- /**
- * Same as the function above, only for one @p solution vector.
- */
- template <int dim, int spacedim, typename VectorType>
- void
- fourier_coefficient_decay(
- const hp::DoFHandler<dim, spacedim> &dof_handler,
- const VectorType & solution,
- Vector<float> & smoothness_indicators,
- const VectorTools::NormType regression_strategy = VectorTools::Linfty_norm);
+ /**
+ * Quadrature collection for the default configuration of the Fourier series
+ * expansion FESeries::Fourier.
+ *
+ * We use a 4-point Gaussian quarature iterated in each dimension by the
+ * maximal wave number, which is the number of modes decresed by one since
+ * we start with $k = 0$.
+ */
+ template <int dim, int spacedim>
+ hp::QCollection<dim>
+ default_quadrature_collection(
+ const hp::FECollection<dim, spacedim> &fe_collection);
+ } // namespace Fourier
} // namespace SmoothnessEstimator
+++ /dev/null
-// ---------------------------------------------------------------------
-//
-// Copyright (C) 1998 - 2018 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_smoothness_estimator_templates_h
-#define dealii_smoothness_estimator_templates_h
-
-#include <deal.II/base/quadrature.h>
-
-#include <deal.II/fe/fe_series.h>
-
-#include <deal.II/hp/q_collection.h>
-
-#include <deal.II/numerics/smoothness_estimator.h>
-
-#include <algorithm>
-#include <cmath>
-#include <functional>
-#include <utility>
-
-
-DEAL_II_NAMESPACE_OPEN
-
-
-namespace SmoothnessEstimator
-{
- namespace
- {
- /**
- * Resizes @p coeff to @p N in each dimension.
- */
- template <int dim, typename CoefficientType>
- void
- resize(Table<dim, CoefficientType> &coeff, const unsigned int N)
- {
- TableIndices<dim> size;
- for (unsigned int d = 0; d < dim; d++)
- size[d] = N;
- coeff.reinit(size);
- }
-
-
-
- /**
- * we will need to take the maximum
- * absolute value of fourier coefficients which correspond to $k$-vector
- * $|{\bf k}|= const$. To filter the coefficients Table we
- * will use the FESeries::process_coefficients() which requires a predicate
- * to be specified. The predicate should operate on TableIndices and return
- * a pair of <code>bool</code> and <code>unsigned int</code>. The latter
- * is the value of the map from TableIndicies to unsigned int. It is
- * used to define subsets of coefficients from which we search for the one
- * with highest absolute value, i.e. $l^\infty$-norm. The <code>bool</code>
- * parameter defines which indices should be used in processing. In the
- * current case we are interested in coefficients which correspond to
- * $0 < i*i+j*j < N*N$ and $0 < i*i+j*j+k*k < N*N$ in 2D and 3D,
- * respectively.
- */
- template <int dim>
- std::pair<bool, unsigned int>
- predicate(const TableIndices<dim> &ind, const unsigned int max_degree)
- {
- unsigned int v = 0;
- for (unsigned int i = 0; i < dim; i++)
- v += ind[i] * ind[i];
- if (v > 0 && v < max_degree * max_degree)
- return std::make_pair(true, v);
- else
- return std::make_pair(false, v);
- }
- } // namespace
-
-
- template <int dim, int spacedim, typename VectorType>
- void
- legendre_coefficient_decay(
- FESeries::Legendre<dim, spacedim> & fe_legendre,
- const hp::DoFHandler<dim, spacedim> & dof_handler,
- const std::vector<const VectorType *> &all_solutions,
- const std::vector<Vector<float> *> & all_smoothness_indicators,
- const std::function<void(std::vector<bool> &flags)> coefficients_predicate,
- const double smallest_abs_coefficient)
- {
- Assert(smallest_abs_coefficient >= 0.,
- ExcMessage("smallest_abs_coefficient should be non-negative."));
-
- using number = typename VectorType::value_type;
- using number_coeff =
- typename FESeries::Legendre<dim, spacedim>::CoefficientType;
-
- AssertDimension(all_solutions.size(), all_smoothness_indicators.size());
-
- for (auto &smoothness_indicator : all_smoothness_indicators)
- smoothness_indicator->reinit(
- dof_handler.get_triangulation().n_active_cells());
-
- Table<dim, number_coeff> expansion_coefficients;
- resize(expansion_coefficients, fe_legendre.get_size_in_each_direction());
-
- Vector<number> local_dof_values;
-
- // auxiliary vector to do linear regression
- std::vector<number_coeff> x;
- std::vector<number_coeff> y;
-
- x.reserve(dof_handler.get_fe_collection().max_degree());
- y.reserve(dof_handler.get_fe_collection().max_degree());
-
- // precalculate predicates for each degree:
- std::vector<std::vector<bool>> predicates(
- dof_handler.get_fe_collection().max_degree());
- for (unsigned int p = 1; p <= dof_handler.get_fe_collection().max_degree();
- ++p)
- {
- auto &pred = predicates[p - 1];
- // we have p+1 coefficients for degree p
- pred.resize(p + 1);
- coefficients_predicate(pred);
- }
-
- for (auto &cell : dof_handler.active_cell_iterators())
- if (cell->is_locally_owned())
- {
- local_dof_values.reinit(cell->get_fe().dofs_per_cell);
-
- const unsigned int pe = cell->get_fe().degree;
-
- Assert(pe > 0, ExcInternalError());
- const auto &pred = predicates[pe - 1];
-
- // since we use coefficients with indices [1,pe] in each direction,
- // the number of coefficients we need to calculate is at least N=pe+1
- AssertIndexRange(pe, fe_legendre.get_size_in_each_direction());
-
- auto solution_it = all_solutions.cbegin();
- auto smoothness_indicators_it = all_smoothness_indicators.begin();
- for (; solution_it != all_solutions.cend();
- ++solution_it, ++smoothness_indicators_it)
- {
- cell->get_dof_values(*(*solution_it), local_dof_values);
- fe_legendre.calculate(local_dof_values,
- cell->active_fe_index(),
- expansion_coefficients);
-
- // choose the smallest decay of coefficients in each direction,
- // i.e. the maximum decay slope k_v
- number_coeff k_v = -std::numeric_limits<number_coeff>::max();
- for (unsigned int d = 0; d < dim; d++)
- {
- x.resize(0);
- y.resize(0);
-
- // will use all non-zero coefficients allowed by the predicate
- // function
- Assert(pred.size() == pe + 1, ExcInternalError());
- for (unsigned int i = 0; i <= pe; i++)
- if (pred[i])
- {
- TableIndices<dim> ind;
- ind[d] = i;
- const number_coeff coeff_abs =
- std::abs(expansion_coefficients(ind));
-
- if (coeff_abs > smallest_abs_coefficient)
- {
- y.push_back(std::log(coeff_abs));
- x.push_back(i);
- }
- }
-
- // in case we don't have enough non-zero coefficient to fit,
- // skip this direction
- if (x.size() < 2)
- continue;
-
- const std::pair<number_coeff, number_coeff> fit =
- FESeries::linear_regression(x, y);
-
- // decay corresponds to negative slope
- // take the lesser negative slope along each direction
- k_v = std::max(k_v, fit.first);
- }
-
- (*(*smoothness_indicators_it))(cell->active_cell_index()) =
- std::exp(k_v);
- }
- }
- }
-
-
-
- template <int dim, int spacedim, typename VectorType>
- void
- legendre_coefficient_decay(
- FESeries::Legendre<dim, spacedim> & fe_legendre,
- const hp::DoFHandler<dim, spacedim> & dof_handler,
- const VectorType & solution,
- Vector<float> & smoothness_indicators,
- const std::function<void(std::vector<bool> &flags)> coefficients_predicate,
- const double smallest_abs_coefficient)
- {
- const std::vector<const VectorType *> all_solutions(1, &solution);
- const std::vector<Vector<float> *> all_smoothness_indicators(
- 1, &smoothness_indicators);
-
- legendre_coefficient_decay(fe_legendre,
- dof_handler,
- all_solutions,
- all_smoothness_indicators,
- coefficients_predicate,
- smallest_abs_coefficient);
- }
-
-
-
- template <int dim, int spacedim, typename VectorType>
- void
- legendre_coefficient_decay(
- const hp::DoFHandler<dim, spacedim> & dof_handler,
- const VectorType & solution,
- Vector<float> & smoothness_indicators,
- const std::function<void(std::vector<bool> &flags)> coefficients_predicate,
- const double smallest_abs_coefficient)
- {
- const unsigned int max_degree =
- dof_handler.get_fe_collection().max_degree();
-
- // We initialize a FESeries::Legendre expansion object object which will be
- // used to calculate the expansion coefficients. In addition to the
- // hp::FECollection, we need to provide quadrature rules hp::QCollection for
- // integration on the reference cell.
- // We will need to assemble the expansion matrices for each of the finite
- // elements we deal with, i.e. the matrices F_k,j. We have to do that for
- // each of the finite elements in use. To that end we need a quadrature
- // rule. As a default, we use the same quadrature formula for each finite
- // element, namely one that is obtained by iterating a 2-point Gauss formula
- // as many times as the maximal polynomial degree.
- QGauss<1> base_quadrature(2);
- QIterated<dim> quadrature(base_quadrature, max_degree);
-
- hp::QCollection<dim> expansion_q_collection;
- for (unsigned int i = 0; i < dof_handler.get_fe_collection().size(); ++i)
- expansion_q_collection.push_back(quadrature);
-
- FESeries::Legendre<dim, spacedim> legendre(max_degree + 1,
- dof_handler.get_fe_collection(),
- expansion_q_collection);
-
- legendre_coefficient_decay(legendre,
- dof_handler,
- solution,
- smoothness_indicators,
- coefficients_predicate,
- smallest_abs_coefficient);
- }
-
-
-
- template <int dim, int spacedim, typename VectorType>
- void
- fourier_coefficient_decay(
- FESeries::Fourier<dim, spacedim> & fe_series,
- const hp::DoFHandler<dim, spacedim> & dof_handler,
- const std::vector<const VectorType *> &all_solutions,
- const std::vector<Vector<float> *> & all_smoothness_indicators,
- const VectorTools::NormType regression_strategy)
- {
- using number = typename VectorType::value_type;
- using number_coeff =
- typename FESeries::Fourier<dim, spacedim>::CoefficientType;
-
- AssertDimension(all_solutions.size(), all_smoothness_indicators.size());
-
- for (auto &smoothness_indicator : all_smoothness_indicators)
- smoothness_indicator->reinit(
- dof_handler.get_triangulation().n_active_cells());
-
- const unsigned int max_degree =
- dof_handler.get_fe_collection().max_degree();
-
- Table<dim, number_coeff> expansion_coefficients;
- resize(expansion_coefficients, max_degree);
-
- Vector<number> local_dof_values;
- std::vector<double> ln_k;
- std::pair<std::vector<unsigned int>, std::vector<double>> res;
- for (auto &cell : dof_handler.active_cell_iterators())
- if (cell->is_locally_owned())
- {
- local_dof_values.reinit(cell->get_fe().dofs_per_cell);
-
- auto solution_it = all_solutions.cbegin();
- auto smoothness_indicators_it = all_smoothness_indicators.begin();
- for (; solution_it != all_solutions.cend();
- ++solution_it, ++smoothness_indicators_it)
- {
- // Inside the loop, we first need to get the values of the local
- // degrees of freedom and then need to compute the series
- // expansion by multiplying this vector with the matrix ${\cal F}$
- // corresponding to this finite element.
- cell->get_dof_values(*(*solution_it), local_dof_values);
-
- fe_series.calculate(local_dof_values,
- cell->active_fe_index(),
- expansion_coefficients);
-
- // We fit our exponential decay of expansion coefficients to the
- // provided regression_strategy on each possible value of |k|. To
- // this end, we use FESeries::process_coefficients() to rework
- // coefficients into the desired format.
- res = FESeries::process_coefficients<dim>(
- expansion_coefficients,
- std::bind(&predicate<dim>, std::placeholders::_1, max_degree),
- regression_strategy);
-
- Assert(res.first.size() == res.second.size(), ExcInternalError());
-
- // Prepare linear equation for the logarithmic least squares fit.
- //
- // First, calculate ln(|k|). This vector will be the same for all
- // the cells so we can calculate ln(|k|) only once.
- //
- // For Fourier expansion, this translates to
- // ln(2*pi*sqrt(predicate)) = ln(2*pi) + 0.5*ln(predicate). Since
- // we are just interested in a linear regression later, we omit
- // the ln(2*pi) factor.
- // For Legendre expansion, this translates to
- // 0.5*ln(predicate) as well, without the pi factor.
- if (ln_k.empty())
- {
- ln_k.resize(res.first.size());
- for (unsigned int f = 0; f < res.first.size(); ++f)
- ln_k[f] = 0.5 * std::log((double)res.first[f]);
- }
-
- // Second, calculate ln(U_k).
- for (auto &residual_element : res.second)
- residual_element = std::log(residual_element);
-
- // Last, do the linear regression.
- const auto fit = FESeries::linear_regression(ln_k, res.second);
-
- // Compute the Sobolev index s=mu-dim/2 and store it in the vector
- // of estimated values for each cell.
- (*(*smoothness_indicators_it))(cell->active_cell_index()) =
- (float)(-fit.first - .5 * dim);
- }
- }
- }
-
-
-
- template <int dim, int spacedim, typename VectorType>
- void
- fourier_coefficient_decay(FESeries::Fourier<dim, spacedim> & fe_series,
- const hp::DoFHandler<dim, spacedim> &dof_handler,
- const VectorType & solution,
- Vector<float> & smoothness_indicators,
- const VectorTools::NormType regression_strategy)
- {
- const std::vector<const VectorType *> all_solutions(1, &solution);
- const std::vector<Vector<float> *> all_smoothness_indicators(
- 1, &smoothness_indicators);
-
- fourier_coefficient_decay(fe_series,
- dof_handler,
- all_solutions,
- all_smoothness_indicators,
- regression_strategy);
- }
-
-
-
- template <int dim, int spacedim, typename VectorType>
- void
- fourier_coefficient_decay(
- const hp::DoFHandler<dim, spacedim> & dof_handler,
- const std::vector<const VectorType *> &all_solutions,
- const std::vector<Vector<float> *> & all_smoothness_indicators,
- const VectorTools::NormType regression_strategy)
- {
- const unsigned int max_degree =
- dof_handler.get_fe_collection().max_degree();
-
- // We initialize a series expansion object object which will be used to
- // calculate the expansion coefficients. In addition to the
- // hp::FECollection, we need to provide quadrature rules hp::QCollection for
- // integration on the reference cell.
- // We will need to assemble the expansion matrices for each of the finite
- // elements we deal with, i.e. the matrices F_k,j. We have to do that for
- // each of the finite elements in use. To that end we need a quadrature
- // rule. As a default, we use the same quadrature formula for each finite
- // element, namely one that is obtained by iterating a 2-point Gauss formula
- // as many times as the maximal exponent we use for the term exp(ikx).
- QGauss<1> base_quadrature(2);
- QIterated<dim> quadrature(base_quadrature, max_degree);
-
- hp::QCollection<dim> expansion_q_collection;
- for (unsigned int i = 0; i < dof_handler.get_fe_collection().size(); ++i)
- expansion_q_collection.push_back(quadrature);
-
- // The FESeries::Fourier class' constructor first parameter $N$ defines the
- // number of coefficients in 1D with the total number of coefficients being
- // $N^{dim}$.
- FESeries::Fourier<dim, spacedim> fe_series(max_degree,
- dof_handler.get_fe_collection(),
- expansion_q_collection);
-
- fourier_coefficient_decay(fe_series,
- dof_handler,
- all_solutions,
- all_smoothness_indicators,
- regression_strategy);
- }
-
-
-
- template <int dim, int spacedim, typename VectorType>
- void
- fourier_coefficient_decay(const hp::DoFHandler<dim, spacedim> &dof_handler,
- const VectorType & solution,
- Vector<float> & smoothness_indicators,
- const VectorTools::NormType regression_strategy)
- {
- const std::vector<const VectorType *> all_solutions(1, &solution);
- const std::vector<Vector<float> *> all_smoothness_indicators(
- 1, &smoothness_indicators);
-
- fourier_coefficient_decay(dof_handler,
- all_solutions,
- all_smoothness_indicators,
- regression_strategy);
- }
-} // namespace SmoothnessEstimator
-
-
-DEAL_II_NAMESPACE_CLOSE
-
-#endif
template <int spacedim>
void
ensure_existence(
+ const std::vector<unsigned int> & n_coefficients_per_direction,
const hp::FECollection<1, spacedim> & fe_collection,
const hp::QCollection<1> & q_collection,
const Table<1, Tensor<1, 1>> & k_vectors,
if (fourier_transform_matrices[fe].m() == 0)
{
- fourier_transform_matrices[fe].reinit(k_vectors.n_elements(),
+ fourier_transform_matrices[fe].reinit(n_coefficients_per_direction[fe],
fe_collection[fe].dofs_per_cell);
- for (unsigned int k = 0; k < k_vectors.size(0); ++k)
+ for (unsigned int k = 0; k < n_coefficients_per_direction[fe]; ++k)
for (unsigned int j = 0; j < fe_collection[fe].dofs_per_cell; ++j)
fourier_transform_matrices[fe](k, j) =
integrate(fe_collection[fe], q_collection[fe], k_vectors(k), j);
template <int spacedim>
void
ensure_existence(
+ const std::vector<unsigned int> & n_coefficients_per_direction,
const hp::FECollection<2, spacedim> & fe_collection,
const hp::QCollection<2> & q_collection,
const Table<2, Tensor<1, 2>> & k_vectors,
if (fourier_transform_matrices[fe].m() == 0)
{
- fourier_transform_matrices[fe].reinit(k_vectors.n_elements(),
- fe_collection[fe].dofs_per_cell);
+ fourier_transform_matrices[fe].reinit(
+ Utilities::fixed_power<2>(n_coefficients_per_direction[fe]),
+ fe_collection[fe].dofs_per_cell);
unsigned int k = 0;
- for (unsigned int k1 = 0; k1 < k_vectors.size(0); ++k1)
- for (unsigned int k2 = 0; k2 < k_vectors.size(1); ++k2, k++)
+ for (unsigned int k1 = 0; k1 < n_coefficients_per_direction[fe]; ++k1)
+ for (unsigned int k2 = 0; k2 < n_coefficients_per_direction[fe];
+ ++k2, ++k)
for (unsigned int j = 0; j < fe_collection[fe].dofs_per_cell; ++j)
fourier_transform_matrices[fe](k, j) = integrate(
fe_collection[fe], q_collection[fe], k_vectors(k1, k2), j);
template <int spacedim>
void
ensure_existence(
+ const std::vector<unsigned int> & n_coefficients_per_direction,
const hp::FECollection<3, spacedim> & fe_collection,
const hp::QCollection<3> & q_collection,
const Table<3, Tensor<1, 3>> & k_vectors,
if (fourier_transform_matrices[fe].m() == 0)
{
- fourier_transform_matrices[fe].reinit(k_vectors.n_elements(),
- fe_collection[fe].dofs_per_cell);
+ fourier_transform_matrices[fe].reinit(
+ Utilities::fixed_power<3>(n_coefficients_per_direction[fe]),
+ fe_collection[fe].dofs_per_cell);
unsigned int k = 0;
- for (unsigned int k1 = 0; k1 < k_vectors.size(0); ++k1)
- for (unsigned int k2 = 0; k2 < k_vectors.size(1); ++k2)
- for (unsigned int k3 = 0; k3 < k_vectors.size(2); ++k3, k++)
+ for (unsigned int k1 = 0; k1 < n_coefficients_per_direction[fe]; ++k1)
+ for (unsigned int k2 = 0; k2 < n_coefficients_per_direction[fe]; ++k2)
+ for (unsigned int k3 = 0; k3 < n_coefficients_per_direction[fe];
+ ++k3, ++k)
for (unsigned int j = 0; j < fe_collection[fe].dofs_per_cell; ++j)
fourier_transform_matrices[fe](k, j) =
integrate(fe_collection[fe],
{
template <int dim, int spacedim>
Fourier<dim, spacedim>::Fourier(
- const unsigned int N,
+ const std::vector<unsigned int> & n_coefficients_per_direction,
const hp::FECollection<dim, spacedim> &fe_collection,
const hp::QCollection<dim> & q_collection)
- : fe_collection(&fe_collection)
+ : n_coefficients_per_direction(n_coefficients_per_direction)
+ , fe_collection(&fe_collection)
, q_collection(&q_collection)
, fourier_transform_matrices(fe_collection.size())
{
- set_k_vectors(k_vectors, N);
- unrolled_coefficients.resize(k_vectors.n_elements());
+ Assert(n_coefficients_per_direction.size() == fe_collection.size() &&
+ n_coefficients_per_direction.size() == q_collection.size(),
+ ExcMessage("All parameters are supposed to have the same size."));
+
+ const unsigned int max_n_coefficients_per_direction =
+ *std::max_element(n_coefficients_per_direction.cbegin(),
+ n_coefficients_per_direction.cend());
+ set_k_vectors(k_vectors, max_n_coefficients_per_direction);
+
+ // reserve sufficient memory
+ unrolled_coefficients.reserve(k_vectors.n_elements());
}
+ template <int dim, int spacedim>
+ Fourier<dim, spacedim>::Fourier(
+ const unsigned int n_coefficients_per_direction,
+ const hp::FECollection<dim, spacedim> &fe_collection,
+ const hp::QCollection<dim> & q_collection)
+ : Fourier<dim, spacedim>(
+ std::vector<unsigned int>(fe_collection.size(),
+ n_coefficients_per_direction),
+ fe_collection,
+ q_collection)
+ {}
+
+
+
template <int dim, int spacedim>
inline bool
Fourier<dim, spacedim>::
operator==(const Fourier<dim, spacedim> &fourier) const
{
- return ((*fe_collection == *(fourier.fe_collection)) &&
- (*q_collection == *(fourier.q_collection)) &&
- (k_vectors == fourier.k_vectors) &&
- (fourier_transform_matrices == fourier.fourier_transform_matrices));
+ return (
+ (n_coefficients_per_direction == fourier.n_coefficients_per_direction) &&
+ (*fe_collection == *(fourier.fe_collection)) &&
+ (*q_collection == *(fourier.q_collection)) &&
+ (k_vectors == fourier.k_vectors) &&
+ (fourier_transform_matrices == fourier.fourier_transform_matrices));
}
Threads::TaskGroup<> task_group;
for (unsigned int fe = 0; fe < fe_collection->size(); ++fe)
task_group += Threads::new_task([&, fe]() {
- ensure_existence(*fe_collection,
+ ensure_existence(n_coefficients_per_direction,
+ *fe_collection,
*q_collection,
k_vectors,
fe,
+ template <int dim, int spacedim>
+ void
+ Fourier<dim, spacedim>::initialize(
+ const std::vector<unsigned int> & n_coefficients_per_direction,
+ const hp::FECollection<dim, spacedim> &fe_collection,
+ const hp::QCollection<dim> & q_collection)
+ {
+ Assert(n_coefficients_per_direction.size() == fe_collection.size() &&
+ n_coefficients_per_direction.size() == q_collection.size(),
+ ExcMessage("All parameters are supposed to have the same size."));
+
+ // set members
+ this->n_coefficients_per_direction = n_coefficients_per_direction;
+ this->fe_collection =
+ SmartPointer<const hp::FECollection<dim, spacedim>>(&fe_collection);
+ this->q_collection =
+ SmartPointer<const hp::QCollection<dim>>(&q_collection);
+
+ // clean up auxiliary members
+ k_vectors.reset_values();
+ const unsigned int max_n_coefficients_per_direction =
+ *std::max_element(n_coefficients_per_direction.cbegin(),
+ n_coefficients_per_direction.cend());
+ set_k_vectors(k_vectors, max_n_coefficients_per_direction);
+
+ fourier_transform_matrices.clear();
+ fourier_transform_matrices.resize(fe_collection.size());
+
+ unrolled_coefficients.clear();
+ // reserve sufficient memory
+ unrolled_coefficients.reserve(k_vectors.n_elements());
+ }
+
+
+
+ template <int dim, int spacedim>
+ unsigned int
+ Fourier<dim, spacedim>::get_n_coefficients_per_direction(
+ const unsigned int index) const
+ {
+ return n_coefficients_per_direction[index];
+ }
+
+
+
template <int dim, int spacedim>
template <typename Number>
void
const unsigned int cell_active_fe_index,
Table<dim, CoefficientType> &fourier_coefficients)
{
- ensure_existence(*fe_collection,
+ Assert(fe_collection != nullptr && q_collection != nullptr,
+ ExcMessage("Initialize this FESeries object first!"));
+ for (unsigned int d = 0; d < dim; ++d)
+ AssertDimension(fourier_coefficients.size(d),
+ n_coefficients_per_direction[cell_active_fe_index]);
+
+ ensure_existence(n_coefficients_per_direction,
+ *fe_collection,
*q_collection,
k_vectors,
cell_active_fe_index,
const FullMatrix<CoefficientType> &matrix =
fourier_transform_matrices[cell_active_fe_index];
+ unrolled_coefficients.resize(Utilities::fixed_power<dim>(
+ n_coefficients_per_direction[cell_active_fe_index]));
std::fill(unrolled_coefficients.begin(),
unrolled_coefficients.end(),
CoefficientType(0.));
*/
template <int spacedim>
void
- ensure_existence(const hp::FECollection<1, spacedim> &fe_collection,
- const hp::QCollection<1> & q_collection,
- const unsigned int N,
- const unsigned int fe,
- std::vector<FullMatrix<double>> &legendre_transform_matrices)
+ ensure_existence(
+ const std::vector<unsigned int> & n_coefficients_per_direction,
+ const hp::FECollection<1, spacedim> &fe_collection,
+ const hp::QCollection<1> & q_collection,
+ const unsigned int fe,
+ std::vector<FullMatrix<double>> & legendre_transform_matrices)
{
AssertIndexRange(fe, fe_collection.size());
if (legendre_transform_matrices[fe].m() == 0)
{
- legendre_transform_matrices[fe].reinit(N,
+ legendre_transform_matrices[fe].reinit(n_coefficients_per_direction[fe],
fe_collection[fe].dofs_per_cell);
- for (unsigned int k = 0; k < N; ++k)
+ for (unsigned int k = 0; k < n_coefficients_per_direction[fe]; ++k)
for (unsigned int j = 0; j < fe_collection[fe].dofs_per_cell; ++j)
legendre_transform_matrices[fe](k, j) = integrate(
fe_collection[fe], q_collection[fe], TableIndices<1>(k), j);
template <int spacedim>
void
- ensure_existence(const hp::FECollection<2, spacedim> &fe_collection,
- const hp::QCollection<2> & q_collection,
- const unsigned int N,
- const unsigned int fe,
- std::vector<FullMatrix<double>> &legendre_transform_matrices)
+ ensure_existence(
+ const std::vector<unsigned int> & n_coefficients_per_direction,
+ const hp::FECollection<2, spacedim> &fe_collection,
+ const hp::QCollection<2> & q_collection,
+ const unsigned int fe,
+ std::vector<FullMatrix<double>> & legendre_transform_matrices)
{
AssertIndexRange(fe, fe_collection.size());
if (legendre_transform_matrices[fe].m() == 0)
{
- legendre_transform_matrices[fe].reinit(N * N,
- fe_collection[fe].dofs_per_cell);
+ legendre_transform_matrices[fe].reinit(
+ Utilities::fixed_power<2>(n_coefficients_per_direction[fe]),
+ fe_collection[fe].dofs_per_cell);
unsigned int k = 0;
- for (unsigned int k1 = 0; k1 < N; ++k1)
- for (unsigned int k2 = 0; k2 < N; ++k2, k++)
+ for (unsigned int k1 = 0; k1 < n_coefficients_per_direction[fe]; ++k1)
+ for (unsigned int k2 = 0; k2 < n_coefficients_per_direction[fe];
+ ++k2, k++)
for (unsigned int j = 0; j < fe_collection[fe].dofs_per_cell; ++j)
legendre_transform_matrices[fe](k, j) =
integrate(fe_collection[fe],
template <int spacedim>
void
- ensure_existence(const hp::FECollection<3, spacedim> &fe_collection,
- const hp::QCollection<3> & q_collection,
- const unsigned int N,
- const unsigned int fe,
- std::vector<FullMatrix<double>> &legendre_transform_matrices)
+ ensure_existence(
+ const std::vector<unsigned int> & n_coefficients_per_direction,
+ const hp::FECollection<3, spacedim> &fe_collection,
+ const hp::QCollection<3> & q_collection,
+ const unsigned int fe,
+ std::vector<FullMatrix<double>> & legendre_transform_matrices)
{
AssertIndexRange(fe, fe_collection.size());
if (legendre_transform_matrices[fe].m() == 0)
{
- legendre_transform_matrices[fe].reinit(N * N * N,
- fe_collection[fe].dofs_per_cell);
+ legendre_transform_matrices[fe].reinit(
+ Utilities::fixed_power<3>(n_coefficients_per_direction[fe]),
+ fe_collection[fe].dofs_per_cell);
unsigned int k = 0;
- for (unsigned int k1 = 0; k1 < N; ++k1)
- for (unsigned int k2 = 0; k2 < N; ++k2)
- for (unsigned int k3 = 0; k3 < N; ++k3, k++)
+ for (unsigned int k1 = 0; k1 < n_coefficients_per_direction[fe]; ++k1)
+ for (unsigned int k2 = 0; k2 < n_coefficients_per_direction[fe]; ++k2)
+ for (unsigned int k3 = 0; k3 < n_coefficients_per_direction[fe];
+ ++k3, k++)
for (unsigned int j = 0; j < fe_collection[fe].dofs_per_cell; ++j)
legendre_transform_matrices[fe](k, j) =
integrate(fe_collection[fe],
{
template <int dim, int spacedim>
Legendre<dim, spacedim>::Legendre(
- const unsigned int size_in_each_direction,
+ const std::vector<unsigned int> & n_coefficients_per_direction,
const hp::FECollection<dim, spacedim> &fe_collection,
const hp::QCollection<dim> & q_collection)
- : N(size_in_each_direction)
+ : n_coefficients_per_direction(n_coefficients_per_direction)
, fe_collection(&fe_collection)
, q_collection(&q_collection)
, legendre_transform_matrices(fe_collection.size())
- , unrolled_coefficients(Utilities::fixed_power<dim>(N), 0.)
+ {
+ Assert(n_coefficients_per_direction.size() == fe_collection.size() &&
+ n_coefficients_per_direction.size() == q_collection.size(),
+ ExcMessage("All parameters are supposed to have the same size."));
+
+ // reserve sufficient memory
+ const unsigned int max_n_coefficients_per_direction =
+ *std::max_element(n_coefficients_per_direction.cbegin(),
+ n_coefficients_per_direction.cend());
+ unrolled_coefficients.reserve(
+ Utilities::fixed_power<dim>(max_n_coefficients_per_direction));
+ }
+
+
+
+ template <int dim, int spacedim>
+ Legendre<dim, spacedim>::Legendre(
+ const unsigned int n_coefficients_per_direction,
+ const hp::FECollection<dim, spacedim> &fe_collection,
+ const hp::QCollection<dim> & q_collection)
+ : Legendre<dim, spacedim>(
+ std::vector<unsigned int>(fe_collection.size(),
+ n_coefficients_per_direction),
+ fe_collection,
+ q_collection)
{}
operator==(const Legendre<dim, spacedim> &legendre) const
{
return (
- (N == legendre.N) && (*fe_collection == *(legendre.fe_collection)) &&
+ (n_coefficients_per_direction == legendre.n_coefficients_per_direction) &&
+ (*fe_collection == *(legendre.fe_collection)) &&
(*q_collection == *(legendre.q_collection)) &&
(legendre_transform_matrices == legendre.legendre_transform_matrices));
}
Threads::TaskGroup<> task_group;
for (unsigned int fe = 0; fe < fe_collection->size(); ++fe)
task_group += Threads::new_task([&, fe]() {
- ensure_existence(
- *fe_collection, *q_collection, N, fe, legendre_transform_matrices);
+ ensure_existence(n_coefficients_per_direction,
+ *fe_collection,
+ *q_collection,
+ fe,
+ legendre_transform_matrices);
});
task_group.join_all();
+ template <int dim, int spacedim>
+ void
+ Legendre<dim, spacedim>::initialize(
+ const std::vector<unsigned int> & n_coefficients_per_direction,
+ const hp::FECollection<dim, spacedim> &fe_collection,
+ const hp::QCollection<dim> & q_collection)
+ {
+ Assert(n_coefficients_per_direction.size() == fe_collection.size() &&
+ n_coefficients_per_direction.size() == q_collection.size(),
+ ExcMessage("All parameters are supposed to have the same size."));
+
+ // set members
+ this->n_coefficients_per_direction = n_coefficients_per_direction;
+ this->fe_collection =
+ SmartPointer<const hp::FECollection<dim, spacedim>>(&fe_collection);
+ this->q_collection =
+ SmartPointer<const hp::QCollection<dim>>(&q_collection);
+
+ // clean up auxiliary members
+ legendre_transform_matrices.clear();
+ legendre_transform_matrices.resize(fe_collection.size());
+
+ unrolled_coefficients.clear();
+ // reserve sufficient memory
+ const unsigned int max_n_coefficients_per_direction =
+ *std::max_element(n_coefficients_per_direction.cbegin(),
+ n_coefficients_per_direction.cend());
+ unrolled_coefficients.reserve(
+ Utilities::fixed_power<dim>(max_n_coefficients_per_direction));
+ }
+
+
+
template <int dim, int spacedim>
unsigned int
- Legendre<dim, spacedim>::get_size_in_each_direction() const
+ Legendre<dim, spacedim>::get_n_coefficients_per_direction(
+ const unsigned int index) const
{
- return N;
+ return n_coefficients_per_direction[index];
}
const unsigned int cell_active_fe_index,
Table<dim, CoefficientType> & legendre_coefficients)
{
+ Assert(fe_collection != nullptr && q_collection != nullptr,
+ ExcMessage("Initialize this FESeries object first!"));
for (unsigned int d = 0; d < dim; ++d)
- AssertDimension(legendre_coefficients.size(d), N);
+ AssertDimension(legendre_coefficients.size(d),
+ n_coefficients_per_direction[cell_active_fe_index]);
- ensure_existence(*fe_collection,
+ ensure_existence(n_coefficients_per_direction,
+ *fe_collection,
*q_collection,
- N,
cell_active_fe_index,
legendre_transform_matrices);
const FullMatrix<CoefficientType> &matrix =
legendre_transform_matrices[cell_active_fe_index];
+ unrolled_coefficients.resize(Utilities::fixed_power<dim>(
+ n_coefficients_per_direction[cell_active_fe_index]));
std::fill(unrolled_coefficients.begin(),
unrolled_coefficients.end(),
CoefficientType(0.));
dealii::GridRefinement::ExcInvalidParameterValue());
// We first have to determine the maximal and minimal values of the
- // criteria of all flagged cells. We start with the minimal and maximal
- // values of all cells, a range within which the minimal and maximal
- // values on cells flagged for refinement must surely lie.
- Number max_criterion_refine =
- *std::min_element(criteria.begin(), criteria.end()),
- min_criterion_refine =
- *std::max_element(criteria.begin(), criteria.end());
+ // criteria of all flagged cells.
+ Number max_criterion_refine = std::numeric_limits<Number>::lowest(),
+ min_criterion_refine = std::numeric_limits<Number>::max();
Number max_criterion_coarsen = max_criterion_refine,
min_criterion_coarsen = min_criterion_refine;
## ---------------------------------------------------------------------
##
-## Copyright (C) 2012 - 2019 by the deal.II authors
+## Copyright (C) 2012 - 2020 by the deal.II authors
##
## This file is part of the deal.II library.
##
// ---------------------------------------------------------------------
//
-// Copyright (C) 1998 - 2018 by the deal.II authors
+// Copyright (C) 2018 - 2020 by the deal.II authors
//
// This file is part of the deal.II library.
//
//
// ---------------------------------------------------------------------
+#include <deal.II/base/quadrature.h>
+#include <deal.II/base/signaling_nan.h>
+
+#include <deal.II/fe/fe_series.h>
+
+#include <deal.II/hp/dof_handler.h>
+#include <deal.II/hp/q_collection.h>
+
#include <deal.II/lac/block_vector.h>
#include <deal.II/lac/la_parallel_block_vector.h>
#include <deal.II/lac/la_parallel_vector.h>
#include <deal.II/lac/trilinos_vector.h>
#include <deal.II/lac/vector.h>
-#include <deal.II/numerics/smoothness_estimator.templates.h>
+#include <deal.II/numerics/smoothness_estimator.h>
+
+#include <algorithm>
+#include <cmath>
+#include <limits>
+#include <utility>
+
DEAL_II_NAMESPACE_OPEN
+
+namespace SmoothnessEstimator
+{
+ namespace
+ {
+ /**
+ * Resizes @p coeff to @p N in each dimension.
+ */
+ template <int dim, typename CoefficientType>
+ void
+ resize(Table<dim, CoefficientType> &coeff, const unsigned int N)
+ {
+ TableIndices<dim> size;
+ for (unsigned int d = 0; d < dim; d++)
+ size[d] = N;
+ coeff.reinit(size);
+ }
+ } // namespace
+
+
+
+ namespace Legendre
+ {
+ namespace
+ {
+ /**
+ * We will need to take the maximum absolute value of Legendre
+ * coefficients which correspond to $k$-vector $|{\bf k}|= const$. To
+ * filter the coefficients Table we will use the
+ * FESeries::process_coefficients() which requires a predicate to be
+ * specified. The predicate should operate on TableIndices and return a
+ * pair of <code>bool</code> and <code>unsigned int</code>. The latter is
+ * the value of the map from TableIndicies to unsigned int. It is used to
+ * define subsets of coefficients from which we search for the one with
+ * highest absolute value, i.e. $l^\infty$-norm. The <code>bool</code>
+ * parameter defines which indices should be used in processing. In the
+ * current case we are interested in coefficients which correspond to $0
+ * <= i+j < N$ and $0 <= i+j+k < N$ in 2D and 3D, respectively.
+ */
+ template <int dim>
+ std::pair<bool, unsigned int>
+ index_sum_less_than_N(const TableIndices<dim> &ind, const unsigned int N)
+ {
+ unsigned int v = 0;
+ for (unsigned int i = 0; i < dim; ++i)
+ v += ind[i];
+
+ return std::make_pair((v < N), v);
+ }
+ } // namespace
+
+
+
+ template <int dim, int spacedim, typename VectorType>
+ void
+ coefficient_decay(FESeries::Legendre<dim, spacedim> & fe_legendre,
+ const hp::DoFHandler<dim, spacedim> &dof_handler,
+ const VectorType & solution,
+ Vector<float> & smoothness_indicators,
+ const VectorTools::NormType regression_strategy,
+ const double smallest_abs_coefficient,
+ const bool only_flagged_cells)
+ {
+ using number = typename VectorType::value_type;
+ using number_coeff =
+ typename FESeries::Legendre<dim, spacedim>::CoefficientType;
+
+ smoothness_indicators.reinit(
+ dof_handler.get_triangulation().n_active_cells());
+
+ unsigned int n_modes;
+ Table<dim, number_coeff> expansion_coefficients;
+
+ Vector<number> local_dof_values;
+ std::vector<double> converted_indices;
+ std::pair<std::vector<unsigned int>, std::vector<double>> res;
+ for (const auto &cell : dof_handler.active_cell_iterators())
+ if (cell->is_locally_owned())
+ {
+ if (!only_flagged_cells || cell->refine_flag_set() ||
+ cell->coarsen_flag_set())
+ {
+ n_modes = fe_legendre.get_n_coefficients_per_direction(
+ cell->active_fe_index());
+ resize(expansion_coefficients, n_modes);
+
+ local_dof_values.reinit(cell->get_fe().dofs_per_cell);
+ cell->get_dof_values(solution, local_dof_values);
+
+ fe_legendre.calculate(local_dof_values,
+ cell->active_fe_index(),
+ expansion_coefficients);
+
+ // We fit our exponential decay of expansion coefficients to the
+ // provided regression_strategy on each possible value of |k|.
+ // To this end, we use FESeries::process_coefficients() to
+ // rework coefficients into the desired format.
+ res = FESeries::process_coefficients<dim>(
+ expansion_coefficients,
+ [n_modes](const TableIndices<dim> &indices) {
+ return index_sum_less_than_N(indices, n_modes);
+ },
+ regression_strategy,
+ smallest_abs_coefficient);
+
+ Assert(res.first.size() == res.second.size(),
+ ExcInternalError());
+
+ // Last, do the linear regression.
+ float regularity = std::numeric_limits<float>::infinity();
+ if (res.first.size() > 1)
+ {
+ // Prepare linear equation for the logarithmic least squares
+ // fit.
+ converted_indices.assign(res.first.begin(),
+ res.first.end());
+
+ for (auto &residual_element : res.second)
+ residual_element = std::log(residual_element);
+
+ const std::pair<double, double> fit =
+ FESeries::linear_regression(converted_indices,
+ res.second);
+ regularity = static_cast<float>(-fit.first);
+ }
+
+ smoothness_indicators(cell->active_cell_index()) = regularity;
+ }
+ else
+ smoothness_indicators(cell->active_cell_index()) =
+ numbers::signaling_nan<float>();
+ }
+ }
+
+
+
+ template <int dim, int spacedim, typename VectorType>
+ void
+ coefficient_decay_per_direction(
+ FESeries::Legendre<dim, spacedim> & fe_legendre,
+ const hp::DoFHandler<dim, spacedim> &dof_handler,
+ const VectorType & solution,
+ Vector<float> & smoothness_indicators,
+ const ComponentMask & coefficients_predicate,
+ const double smallest_abs_coefficient,
+ const bool only_flagged_cells)
+ {
+ Assert(smallest_abs_coefficient >= 0.,
+ ExcMessage("smallest_abs_coefficient should be non-negative."));
+
+ using number = typename VectorType::value_type;
+ using number_coeff =
+ typename FESeries::Legendre<dim, spacedim>::CoefficientType;
+
+ smoothness_indicators.reinit(
+ dof_handler.get_triangulation().n_active_cells());
+
+ unsigned int n_modes;
+ Table<dim, number_coeff> expansion_coefficients;
+ Vector<number> local_dof_values;
+
+ // auxiliary vector to do linear regression
+ const unsigned int max_degree =
+ dof_handler.get_fe_collection().max_degree();
+
+ std::vector<double> x, y;
+ x.reserve(max_degree);
+ y.reserve(max_degree);
+
+ for (const auto &cell : dof_handler.active_cell_iterators())
+ if (cell->is_locally_owned())
+ {
+ if (!only_flagged_cells || cell->refine_flag_set() ||
+ cell->coarsen_flag_set())
+ {
+ n_modes = fe_legendre.get_n_coefficients_per_direction(
+ cell->active_fe_index());
+ resize(expansion_coefficients, n_modes);
+
+ const unsigned int pe = cell->get_fe().degree;
+ Assert(pe > 0, ExcInternalError());
+
+ // since we use coefficients with indices [1,pe] in each
+ // direction, the number of coefficients we need to calculate is
+ // at least N=pe+1
+ AssertIndexRange(pe, n_modes);
+
+ local_dof_values.reinit(cell->get_fe().dofs_per_cell);
+ cell->get_dof_values(solution, local_dof_values);
+
+ fe_legendre.calculate(local_dof_values,
+ cell->active_fe_index(),
+ expansion_coefficients);
+
+ // choose the smallest decay of coefficients in each direction,
+ // i.e. the maximum decay slope k_v as in exp(-k_v)
+ double k_v = std::numeric_limits<double>::infinity();
+ for (unsigned int d = 0; d < dim; ++d)
+ {
+ x.resize(0);
+ y.resize(0);
+
+ // will use all non-zero coefficients allowed by the
+ // predicate function
+ for (unsigned int i = 0; i <= pe; ++i)
+ if (coefficients_predicate[i])
+ {
+ TableIndices<dim> ind;
+ ind[d] = i;
+ const double coeff_abs =
+ std::abs(expansion_coefficients(ind));
+
+ if (coeff_abs > smallest_abs_coefficient)
+ {
+ x.push_back(i);
+ y.push_back(std::log(coeff_abs));
+ }
+ }
+
+ // in case we don't have enough non-zero coefficient to fit,
+ // skip this direction
+ if (x.size() < 2)
+ continue;
+
+ const std::pair<double, double> fit =
+ FESeries::linear_regression(x, y);
+
+ // decay corresponds to negative slope
+ // take the lesser negative slope along each direction
+ k_v = std::min(k_v, -fit.first);
+ }
+
+ smoothness_indicators(cell->active_cell_index()) =
+ static_cast<float>(k_v);
+ }
+ else
+ smoothness_indicators(cell->active_cell_index()) =
+ numbers::signaling_nan<float>();
+ }
+ }
+
+
+
+ template <int dim, int spacedim>
+ std::vector<unsigned int>
+ default_number_of_coefficients_per_direction(
+ const hp::FECollection<dim, spacedim> &fe_collection)
+ {
+ std::vector<unsigned int> n_coefficients_per_direction;
+ for (unsigned int i = 0; i < fe_collection.size(); ++i)
+ n_coefficients_per_direction.push_back(fe_collection[i].degree + 1);
+
+ return n_coefficients_per_direction;
+ }
+
+
+
+ template <int dim, int spacedim>
+ hp::QCollection<dim>
+ default_quadrature_collection(
+ const hp::FECollection<dim, spacedim> &fe_collection)
+ {
+ const std::vector<unsigned int> n_modes =
+ default_number_of_coefficients_per_direction(fe_collection);
+
+ // We initialize a FESeries::Legendre expansion object object which will
+ // be used to calculate the expansion coefficients. In addition to the
+ // hp::FECollection, we need to provide quadrature rules hp::QCollection
+ // for integration on the reference cell.
+ // We will need to assemble the expansion matrices for each of the finite
+ // elements we deal with, i.e. the matrices F_k,j. We have to do that for
+ // each of the finite elements in use. To that end we need a quadrature
+ // rule. As a default, we use the same quadrature formula for each finite
+ // element, namely a Gauss formula that yields exact results for the
+ // highest order Legendre polynomial used.
+
+ // We start with the zeroth Legendre polynomial which is just a constant,
+ // so the highest Legendre polynomial will be of order (n_modes - 1).
+ hp::QCollection<dim> q_collection;
+ for (unsigned int i = 0; i < fe_collection.size(); ++i)
+ {
+ const QGauss<dim> quadrature(n_modes[i]);
+ const QSorted<dim> quadrature_sorted(quadrature);
+ q_collection.push_back(quadrature_sorted);
+ }
+
+ return q_collection;
+ }
+ } // namespace Legendre
+
+
+
+ namespace Fourier
+ {
+ namespace
+ {
+ /**
+ * We will need to take the maximum absolute value of Fourier coefficients
+ * which correspond to $k$-vector $|{\bf k}|= const$. To filter the
+ * coefficients Table we will use the FESeries::process_coefficients()
+ * which requires a predicate to be specified. The predicate should
+ * operate on TableIndices and return a pair of <code>bool</code> and
+ * <code>unsigned int</code>. The latter is the value of the map from
+ * TableIndicies to unsigned int. It is used to define subsets of
+ * coefficients from which we search for the one with highest absolute
+ * value, i.e. $l^\infty$-norm. The <code>bool</code> parameter defines
+ * which indices should be used in processing. In the current case we are
+ * interested in coefficients which correspond to $0 < i^2+j^2 < N^2$ and
+ * $0 < i^2+j^2+k^2 < N^2$ in 2D and 3D, respectively.
+ */
+ template <int dim>
+ std::pair<bool, unsigned int>
+ index_norm_greater_than_zero_and_less_than_N_squared(
+ const TableIndices<dim> &ind,
+ const unsigned int N)
+ {
+ unsigned int v = 0;
+ for (unsigned int i = 0; i < dim; ++i)
+ v += ind[i] * ind[i];
+
+ return std::make_pair((v > 0 && v < N * N), v);
+ }
+ } // namespace
+
+
+
+ template <int dim, int spacedim, typename VectorType>
+ void
+ coefficient_decay(FESeries::Fourier<dim, spacedim> & fe_fourier,
+ const hp::DoFHandler<dim, spacedim> &dof_handler,
+ const VectorType & solution,
+ Vector<float> & smoothness_indicators,
+ const VectorTools::NormType regression_strategy,
+ const double smallest_abs_coefficient,
+ const bool only_flagged_cells)
+ {
+ using number = typename VectorType::value_type;
+ using number_coeff =
+ typename FESeries::Fourier<dim, spacedim>::CoefficientType;
+
+ smoothness_indicators.reinit(
+ dof_handler.get_triangulation().n_active_cells());
+
+ unsigned int n_modes;
+ Table<dim, number_coeff> expansion_coefficients;
+
+ Vector<number> local_dof_values;
+ std::vector<double> ln_k;
+ std::pair<std::vector<unsigned int>, std::vector<double>> res;
+ for (const auto &cell : dof_handler.active_cell_iterators())
+ if (cell->is_locally_owned())
+ {
+ if (!only_flagged_cells || cell->refine_flag_set() ||
+ cell->coarsen_flag_set())
+ {
+ n_modes = fe_fourier.get_n_coefficients_per_direction(
+ cell->active_fe_index());
+ resize(expansion_coefficients, n_modes);
+
+ // Inside the loop, we first need to get the values of the local
+ // degrees of freedom and then need to compute the series
+ // expansion by multiplying this vector with the matrix ${\cal
+ // F}$ corresponding to this finite element.
+ local_dof_values.reinit(cell->get_fe().dofs_per_cell);
+ cell->get_dof_values(solution, local_dof_values);
+
+ fe_fourier.calculate(local_dof_values,
+ cell->active_fe_index(),
+ expansion_coefficients);
+
+ // We fit our exponential decay of expansion coefficients to the
+ // provided regression_strategy on each possible value of |k|.
+ // To this end, we use FESeries::process_coefficients() to
+ // rework coefficients into the desired format.
+ res = FESeries::process_coefficients<dim>(
+ expansion_coefficients,
+ [n_modes](const TableIndices<dim> &indices) {
+ return index_norm_greater_than_zero_and_less_than_N_squared(
+ indices, n_modes);
+ },
+ regression_strategy,
+ smallest_abs_coefficient);
+
+ Assert(res.first.size() == res.second.size(),
+ ExcInternalError());
+
+ // Last, do the linear regression.
+ float regularity = std::numeric_limits<float>::infinity();
+ if (res.first.size() > 1)
+ {
+ // Prepare linear equation for the logarithmic least squares
+ // fit.
+ //
+ // First, calculate ln(|k|).
+ //
+ // For Fourier expansion, this translates to
+ // ln(2*pi*sqrt(predicate)) = ln(2*pi) + 0.5*ln(predicate).
+ // Since we are just interested in the slope of a linear
+ // regression later, we omit the ln(2*pi) factor.
+ ln_k.resize(res.first.size());
+ for (unsigned int f = 0; f < res.first.size(); ++f)
+ ln_k[f] =
+ 0.5 * std::log(static_cast<double>(res.first[f]));
+
+ // Second, calculate ln(U_k).
+ for (auto &residual_element : res.second)
+ residual_element = std::log(residual_element);
+
+ const std::pair<double, double> fit =
+ FESeries::linear_regression(ln_k, res.second);
+ // Compute regularity s = mu - dim/2
+ regularity = static_cast<float>(-fit.first) -
+ ((dim > 1) ? (.5 * dim) : 0);
+ }
+
+ // Store result in the vector of estimated values for each cell.
+ smoothness_indicators(cell->active_cell_index()) = regularity;
+ }
+ else
+ smoothness_indicators(cell->active_cell_index()) =
+ numbers::signaling_nan<float>();
+ }
+ }
+
+
+
+ template <int dim, int spacedim, typename VectorType>
+ void
+ coefficient_decay_per_direction(
+ FESeries::Fourier<dim, spacedim> & fe_fourier,
+ const hp::DoFHandler<dim, spacedim> &dof_handler,
+ const VectorType & solution,
+ Vector<float> & smoothness_indicators,
+ const ComponentMask & coefficients_predicate,
+ const double smallest_abs_coefficient,
+ const bool only_flagged_cells)
+ {
+ Assert(smallest_abs_coefficient >= 0.,
+ ExcMessage("smallest_abs_coefficient should be non-negative."));
+
+ using number = typename VectorType::value_type;
+ using number_coeff =
+ typename FESeries::Fourier<dim, spacedim>::CoefficientType;
+
+ smoothness_indicators.reinit(
+ dof_handler.get_triangulation().n_active_cells());
+
+ unsigned int n_modes;
+ Table<dim, number_coeff> expansion_coefficients;
+ Vector<number> local_dof_values;
+
+ // auxiliary vector to do linear regression
+ const unsigned int max_degree =
+ dof_handler.get_fe_collection().max_degree();
+
+ std::vector<double> x, y;
+ x.reserve(max_degree);
+ y.reserve(max_degree);
+
+ for (const auto &cell : dof_handler.active_cell_iterators())
+ if (cell->is_locally_owned())
+ {
+ if (!only_flagged_cells || cell->refine_flag_set() ||
+ cell->coarsen_flag_set())
+ {
+ n_modes = fe_fourier.get_n_coefficients_per_direction(
+ cell->active_fe_index());
+ resize(expansion_coefficients, n_modes);
+
+ const unsigned int pe = cell->get_fe().degree;
+ Assert(pe > 0, ExcInternalError());
+
+ // since we use coefficients with indices [1,pe] in each
+ // direction, the number of coefficients we need to calculate is
+ // at least N=pe+1
+ AssertIndexRange(pe, n_modes);
+
+ local_dof_values.reinit(cell->get_fe().dofs_per_cell);
+ cell->get_dof_values(solution, local_dof_values);
+
+ fe_fourier.calculate(local_dof_values,
+ cell->active_fe_index(),
+ expansion_coefficients);
+
+ // choose the smallest decay of coefficients in each direction,
+ // i.e. the maximum decay slope k_v as in exp(-k_v)
+ double k_v = std::numeric_limits<double>::infinity();
+ for (unsigned int d = 0; d < dim; ++d)
+ {
+ x.resize(0);
+ y.resize(0);
+
+ // will use all non-zero coefficients allowed by the
+ // predicate function
+ //
+ // skip i=0 because of logarithm
+ for (unsigned int i = 1; i <= pe; ++i)
+ if (coefficients_predicate[i])
+ {
+ TableIndices<dim> ind;
+ ind[d] = i;
+ const double coeff_abs =
+ std::abs(expansion_coefficients(ind));
+
+ if (coeff_abs > smallest_abs_coefficient)
+ {
+ x.push_back(std::log(i));
+ y.push_back(std::log(coeff_abs));
+ }
+ }
+
+ // in case we don't have enough non-zero coefficient to fit,
+ // skip this direction
+ if (x.size() < 2)
+ continue;
+
+ const std::pair<double, double> fit =
+ FESeries::linear_regression(x, y);
+
+ // decay corresponds to negative slope
+ // take the lesser negative slope along each direction
+ k_v = std::min(k_v, -fit.first);
+ }
+
+ smoothness_indicators(cell->active_cell_index()) =
+ static_cast<float>(k_v);
+ }
+ else
+ smoothness_indicators(cell->active_cell_index()) =
+ numbers::signaling_nan<float>();
+ }
+ }
+
+
+
+ template <int dim, int spacedim>
+ std::vector<unsigned int>
+ default_number_of_coefficients_per_direction(
+ const hp::FECollection<dim, spacedim> &fe_collection)
+ {
+ std::vector<unsigned int> n_coefficients_per_direction;
+ for (unsigned int i = 0; i < fe_collection.size(); ++i)
+ n_coefficients_per_direction.push_back(
+ std::max<unsigned int>(3, fe_collection[i].degree + 1));
+
+ return n_coefficients_per_direction;
+ }
+
+
+
+ template <int dim, int spacedim>
+ hp::QCollection<dim>
+ default_quadrature_collection(
+ const hp::FECollection<dim, spacedim> &fe_collection)
+ {
+ const std::vector<unsigned int> n_modes =
+ default_number_of_coefficients_per_direction(fe_collection);
+
+ // We initialize a series expansion object object which will be used to
+ // calculate the expansion coefficients. In addition to the
+ // hp::FECollection, we need to provide quadrature rules hp::QCollection
+ // for integration on the reference cell.
+ // We will need to assemble the expansion matrices for each of the finite
+ // elements we deal with, i.e. the matrices F_k,j. We have to do that for
+ // each of the finite elements in use. To that end we need a quadrature
+ // rule. As a default, we use the same quadrature formula for each finite
+ // element, namely one that is obtained by iterating a 4-point Gauss
+ // formula as many times as the maximal exponent we use for the term
+ // exp(ikx). Since the first mode corresponds to k = 0, the maximal wave
+ // number is k = n_modes - 1.
+ const QGauss<1> base_quadrature(4);
+ hp::QCollection<dim> q_collection;
+ for (unsigned int i = 0; i < fe_collection.size(); ++i)
+ {
+ const QIterated<dim> quadrature(base_quadrature, n_modes[i] - 1);
+ const QSorted<dim> quadrature_sorted(quadrature);
+ q_collection.push_back(quadrature_sorted);
+ }
+
+ return q_collection;
+ }
+ } // namespace Fourier
+} // namespace SmoothnessEstimator
+
+
+// explicit instantiations
#include "smoothness_estimator.inst"
DEAL_II_NAMESPACE_CLOSE
// ---------------------------------------------------------------------
//
-// Copyright (C) 2010 - 2018 by the deal.II authors
+// Copyright (C) 2018 - 2020 by the deal.II authors
//
// This file is part of the deal.II library.
//
for (deal_II_dimension : DIMENSIONS; deal_II_space_dimension : SPACE_DIMENSIONS;
VEC : REAL_VECTOR_TYPES)
{
-#if deal_II_dimension != 1 && deal_II_dimension <= deal_II_space_dimension
- template void SmoothnessEstimator::legendre_coefficient_decay<
+#if deal_II_dimension <= deal_II_space_dimension
+ template void SmoothnessEstimator::Legendre::
+ coefficient_decay<deal_II_dimension, deal_II_space_dimension, VEC>(
+ FESeries::Legendre<deal_II_dimension, deal_II_space_dimension> &,
+ const hp::DoFHandler<deal_II_dimension, deal_II_space_dimension> &,
+ const VEC &,
+ Vector<float> &,
+ const VectorTools::NormType,
+ const double,
+ const bool);
+ template void
+ SmoothnessEstimator::Legendre::coefficient_decay_per_direction<
deal_II_dimension,
deal_II_space_dimension,
VEC>(FESeries::Legendre<deal_II_dimension, deal_II_space_dimension> &,
const hp::DoFHandler<deal_II_dimension, deal_II_space_dimension> &,
- const std::vector<const VEC *> &,
- const std::vector<Vector<float> *> &,
- const std::function<void(std::vector<bool> & flags)>,
- const double);
-
- template void SmoothnessEstimator::legendre_coefficient_decay<
- deal_II_dimension,
- deal_II_space_dimension,
- VEC>(FESeries::Legendre<deal_II_dimension, deal_II_space_dimension> &,
- const hp::DoFHandler<deal_II_dimension, deal_II_space_dimension> &,
- const VEC &,
- Vector<float> &,
- const std::function<void(std::vector<bool> & flags)>,
- const double);
-
- template void SmoothnessEstimator::legendre_coefficient_decay<
- deal_II_dimension,
- deal_II_space_dimension,
- VEC>(const hp::DoFHandler<deal_II_dimension, deal_II_space_dimension> &,
const VEC &,
Vector<float> &,
- const std::function<void(std::vector<bool> & flags)>,
- const double);
+ const ComponentMask &,
+ const double,
+ const bool);
- template void SmoothnessEstimator::fourier_coefficient_decay<
- deal_II_dimension,
- deal_II_space_dimension,
- VEC>(FESeries::Fourier<deal_II_dimension, deal_II_space_dimension> &,
- const hp::DoFHandler<deal_II_dimension, deal_II_space_dimension> &,
- const std::vector<const VEC *> &,
- const std::vector<Vector<float> *> &,
- const VectorTools::NormType);
- template void SmoothnessEstimator::fourier_coefficient_decay<
+ template void SmoothnessEstimator::Fourier::
+ coefficient_decay<deal_II_dimension, deal_II_space_dimension, VEC>(
+ FESeries::Fourier<deal_II_dimension, deal_II_space_dimension> &,
+ const hp::DoFHandler<deal_II_dimension, deal_II_space_dimension> &,
+ const VEC &,
+ Vector<float> &,
+ const VectorTools::NormType,
+ const double,
+ const bool);
+ template void SmoothnessEstimator::Fourier::coefficient_decay_per_direction<
deal_II_dimension,
deal_II_space_dimension,
VEC>(FESeries::Fourier<deal_II_dimension, deal_II_space_dimension> &,
const hp::DoFHandler<deal_II_dimension, deal_II_space_dimension> &,
const VEC &,
Vector<float> &,
- const VectorTools::NormType);
+ const ComponentMask &,
+ const double,
+ const bool);
+#endif
+ }
- template void SmoothnessEstimator::fourier_coefficient_decay<
- deal_II_dimension,
- deal_II_space_dimension,
- VEC>(const hp::DoFHandler<deal_II_dimension, deal_II_space_dimension> &,
- const std::vector<const VEC *> &,
- const std::vector<Vector<float> *> &,
- const VectorTools::NormType);
- template void SmoothnessEstimator::fourier_coefficient_decay<
- deal_II_dimension,
- deal_II_space_dimension,
- VEC>(const hp::DoFHandler<deal_II_dimension, deal_II_space_dimension> &,
- const VEC &,
- Vector<float> &,
- const VectorTools::NormType);
+for (deal_II_dimension : DIMENSIONS; deal_II_space_dimension : SPACE_DIMENSIONS)
+ {
+#if deal_II_dimension <= deal_II_space_dimension
+ template std::vector<unsigned int> SmoothnessEstimator::Legendre::
+ default_number_of_coefficients_per_direction<deal_II_dimension,
+ deal_II_space_dimension>(
+ const hp::FECollection<deal_II_dimension, deal_II_space_dimension> &);
+ template hp::QCollection<deal_II_dimension> SmoothnessEstimator::Legendre::
+ default_quadrature_collection<deal_II_dimension, deal_II_space_dimension>(
+ const hp::FECollection<deal_II_dimension, deal_II_space_dimension> &);
+ template std::vector<unsigned int> SmoothnessEstimator::Fourier::
+ default_number_of_coefficients_per_direction<deal_II_dimension,
+ deal_II_space_dimension>(
+ const hp::FECollection<deal_II_dimension, deal_II_space_dimension> &);
+ template hp::QCollection<deal_II_dimension> SmoothnessEstimator::Fourier::
+ default_quadrature_collection<deal_II_dimension, deal_II_space_dimension>(
+ const hp::FECollection<deal_II_dimension, deal_II_space_dimension> &);
#endif
}
QIterated<dim> quadrature(base_quadrature, N);
q_collection.push_back(quadrature);
- FESeries::Fourier<dim> fourier(N, fe_collection, q_collection);
+ const std::vector<unsigned int> n_coefficients_per_direction(1, N);
+
+ FESeries::Fourier<dim> fourier(n_coefficients_per_direction,
+ fe_collection,
+ q_collection);
Vector<double> local_dof_values(2);
local_dof_values[0] = 0;
VectorTools::interpolate(dof_handler, func, values);
- const unsigned int N = poly_degree + 1;
- FESeries::Legendre<dim> legendre(N, fe_collection, quadrature_formula);
+ const unsigned int N = poly_degree + 1;
+ const std::vector<unsigned int> n_coefficients_per_direction(
+ fe_collection.size(), N);
+ FESeries::Legendre<dim> legendre(n_coefficients_per_direction,
+ fe_collection,
+ quadrature_formula);
const std::vector<double> &coeff_in = func.get_coefficients();
Table<1, double> coeff_out(N);
DEAL::L2_norm
-DEAL::0 : 0
DEAL::1 : 4.12311
DEAL::2 : 9.64365
DEAL::3 : 16.4317
DEAL::5 : 17.8045
DEAL::6 : 15.0000
DEAL::L1_norm
-DEAL::0 : 0
DEAL::1 : 5.00000
DEAL::2 : 15.0000
DEAL::3 : 30.0000
DEAL::5 : 25.0000
DEAL::6 : 15.0000
DEAL::Linfty_norm
-DEAL::0 : 0
DEAL::1 : 4.00000
DEAL::2 : 8.00000
DEAL::3 : 12.0000
DEAL::5 : 14.0000
DEAL::6 : 15.0000
DEAL::mean
-DEAL::0 : 0
DEAL::1 : 2.50000
DEAL::2 : 5.00000
DEAL::3 : 7.50000
Vector<double> values(dof_handler.n_dofs());
VectorTools::interpolate(dof_handler, func, values);
- const unsigned int N = 4;
- FESeries::Legendre<dim> legendre(N, fe_collection, quadrature_formula);
+
+ const unsigned int N = 4;
+ const std::vector<unsigned int> n_coefficients_per_direction(
+ fe_collection.size(), N);
+ FESeries::Legendre<dim> legendre(n_coefficients_per_direction,
+ fe_collection,
+ quadrature_formula);
Table<1, double> coeff_out(N);
Vector<double> local_dof_values;
VectorTools::interpolate(dof_handler, func, values);
- const unsigned int N = poly_degree + 1;
- FESeries::Legendre<dim> legendre(N, fe_collection, quadrature_formula);
+ const unsigned int N = poly_degree + 1;
+ const std::vector<unsigned int> n_coefficients_per_direction(
+ fe_collection.size(), N);
+ FESeries::Legendre<dim> legendre(n_coefficients_per_direction,
+ fe_collection,
+ quadrature_formula);
const Table<dim, double> &coeff_in = func.get_coefficients();
Table<dim, double> coeff_out;
for (unsigned int i = 0; i < fe_collection.size(); i++)
fourier_q_collection.push_back(quadrature);
- FESeries::Fourier<dim> fourier(N, fe_collection, fourier_q_collection);
+ const std::vector<unsigned int> n_coefficients_per_direction(
+ fe_collection.size(), N);
+
+ FESeries::Fourier<dim> fourier(n_coefficients_per_direction,
+ fe_collection,
+ fourier_q_collection);
+
Table<dim, std::complex<double>> fourier_coefficients;
fourier_coefficients.reinit(N, N);
DEAL::L2_norm
-DEAL::0 : 0
DEAL::1 : 4.12311
DEAL::2 : 9.64365
DEAL::3 : 16.4317
DEAL::L1_norm
-DEAL::0 : 0
DEAL::1 : 5.00000
DEAL::2 : 15.0000
DEAL::3 : 30.0000
DEAL::Linfty_norm
-DEAL::0 : 0
DEAL::1 : 4.00000
DEAL::2 : 8.00000
DEAL::3 : 12.0000
DEAL::mean
-DEAL::0 : 0
DEAL::1 : 2.50000
DEAL::2 : 5.00000
DEAL::3 : 7.50000
// ---------------------------------------------------------------------
//
-// Copyright (C) 2018 by the deal.II authors
+// Copyright (C) 2018 - 2020 by the deal.II authors
//
// This file is part of the deal.II library.
//
//
// ---------------------------------------------------------------------
+
+
// base header for hp-FEM test on Laplace equation.
+
#include <deal.II/base/convergence_table.h>
#include <deal.II/base/function.h>
#include <deal.II/base/logstream.h>
#include <deal.II/fe/fe_q.h>
#include <deal.II/fe/fe_q_hierarchical.h>
+#include <deal.II/fe/fe_series.h>
#include <deal.II/fe/fe_system.h>
#include <deal.II/fe/fe_tools.h>
#include <deal.II/fe/fe_values.h>
#include <deal.II/grid/tria_accessor.h>
#include <deal.II/grid/tria_iterator.h>
-#include <deal.II/lac/constraint_matrix.h>
+#include <deal.II/hp/dof_handler.h>
+#include <deal.II/hp/fe_collection.h>
+#include <deal.II/hp/q_collection.h>
+#include <deal.II/hp/refinement.h>
+
+#include <deal.II/lac/affine_constraints.h>
#include <deal.II/lac/dynamic_sparsity_pattern.h>
#include <deal.II/lac/full_matrix.h>
#include <deal.II/lac/precondition.h>
#include "../tests.h"
+
+
/**
* Basic class for Laplace problem
*/
mark_h_cells() = 0;
/**
- * remove h-refinement flag from some cells and populate @p p_cells with
- * iterators to those cells, that shall be p-refined.
+ * remove h-refinement flag from some cells and flag cells for p-refinement
*/
- virtual std::pair<unsigned int, unsigned int>
- substitute_h_for_p(
- std::vector<typename Triangulation<dim>::active_cell_iterator>
- &p_cells) = 0;
+ virtual void
+ substitute_h_for_p() = 0;
void
refine_grid(const unsigned int cycle);
deallog << std::endl;
}
+
+
template <int dim>
Laplace<dim>::~Laplace()
{
dof_handler.clear();
}
+
+
template <int dim>
hp::DoFHandler<dim> &
Laplace<dim>::get_dof_handler()
return dof_handler;
}
+
+
template <int dim>
void
Laplace<dim>::setup_solve_estimate(Vector<float> &output_estimate)
output_estimate = estimated_error_per_cell;
}
+
+
template <int dim>
void
Laplace<dim>::setup_system()
<< std::endl;
}
+
+
template <int dim>
void
Laplace<dim>::assemble()
pcout << " done." << std::endl;
}
+
+
//#define DIRECT
template <int dim>
pcout << " done." << std::endl;
}
+
+
template <int dim>
void
Laplace<dim>::refine_grid(const unsigned int cycle)
mark_h_cells();
// 3.3. Substitute h for p refinement
- std::vector<typename Triangulation<dim>::active_cell_iterator> p_cells;
- hp_number = substitute_h_for_p(p_cells);
+ substitute_h_for_p();
+ // prepare refinement and store number of flagged cells
triangulation.prepare_coarsening_and_refinement();
+ hp_number = {0, 0};
+ for (const auto &cell : dof_handler.active_cell_iterators())
+ {
+ if (cell->refine_flag_set())
+ ++hp_number.first;
+ if (cell->future_fe_index_set())
+ ++hp_number.second;
+ }
+
// 3.4. Solution Transfer
SolutionTransfer<dim, TrilinosWrappers::MPI::Vector, hp::DoFHandler<dim>>
soltrans(dof_handler);
solution_coarse = solution;
soltrans.prepare_for_coarsening_and_refinement(solution_coarse);
- // increase fe_index()
- for (unsigned int i = 0; i < p_cells.size(); i++)
- {
- typename hp::DoFHandler<dim>::active_cell_iterator cell(
- &triangulation, p_cells[i]->level(), p_cells[i]->index(), &dof_handler);
-
- const unsigned int incremented_index = cell->active_fe_index() + 1;
- Assert(incremented_index < fe.size(), ExcInternalError());
- cell->set_active_fe_index(incremented_index);
- }
-
- // 3.5. Refinement
+ // 3.5. h-refinement and p-refinement
triangulation.execute_coarsening_and_refinement();
// FIXME: some hp strategies might need:
soltrans.interpolate(solution_coarse, solution);
}
+
+
template <int dim>
void
Laplace<dim>::calculate_error()
Linfty_error = Utilities::MPI::max(Linfty_error, mpi_communicator);
}
+
+
template <int dim>
void
Laplace<dim>::output_results(int cycle)
<< Linfty_error << sp << total_error << sp << std::endl;
}
+
+
template <int dim>
void
Laplace<dim>::print_errors()
}
}
+
+
template <int dim>
void
Laplace<dim>::run()
+++ /dev/null
-
-DEAL::
-DEAL::0 256 0 0 289 0.267081 1.78414 1.23386 1.29971
-DEAL::1 256 0 56 477 0.0236257 0.743531 0.249786 1.18708
-DEAL::2 268 4 48 801 0.00108376 0.293919 0.0466411 0.361176
-DEAL::3 349 22 32 1730 0.000121568 0.0610935 0.00469929 0.0464975
-DEAL::4 421 18 52 2818 3.08015e-05 0.0204013 0.00315297 0.0197872
-DEAL::5 496 11 74 4632 6.14178e-06 0.00604078 0.000714754 0.00530244
-DEAL::6 622 28 72 7303 2.03947e-06 0.00210333 0.000148863 0.00185557
-DEAL::7 712 17 108 10086 5.30920e-07 0.000744017 7.24474e-05 0.000775517
-DEAL::8 811 20 123 13127 2.97635e-07 0.000379131 3.62580e-05 0.000400767
-DEAL::9 901 15 148 17913 1.18544e-07 0.000194730 1.48736e-05 0.000216587
-DEAL::10 1030 20 161 23692 4.78884e-08 0.000106719 9.54505e-06 0.000117773
// ---------------------------------------------------------------------
//
-// Copyright (C) 2018 by the deal.II authors
+// Copyright (C) 2018 - 2020 by the deal.II authors
//
// This file is part of the deal.II library.
//
//
// ---------------------------------------------------------------------
-// test SmoothnessEstimator::legendre_coefficient_decay() on
-// problem 4 (peak) in Mitchel 2014.
+
+
+// test SmoothnessEstimator::Legendre::coefficient_decay_per_direction()
+// on problem 4 (peak) in Mitchell 2014.
+
#include "laplace.h"
estimate_error();
void
mark_h_cells();
-
- std::pair<unsigned int, unsigned int>
- substitute_h_for_p(
- std::vector<typename Triangulation<dim>::active_cell_iterator> &p_cells);
+ void
+ substitute_h_for_p();
hp::QCollection<dim - 1> quadrature_face;
+
+ std::vector<unsigned int> n_coefficients_per_direction;
+ hp::QCollection<dim> expansion_q_collection;
+ FESeries::Legendre<dim> legendre;
};
template <int dim>
const QIterated<dim> q_iterated(q_trapez, p + 3);
Laplace<dim>::quadrature_infty.push_back(QSorted<dim>(q_iterated));
}
+
+ // after the FECollection has been generated, create a corresponding legendre
+ // series expansion object
+ n_coefficients_per_direction =
+ SmoothnessEstimator::Legendre::default_number_of_coefficients_per_direction(
+ Laplace<dim>::fe);
+ expansion_q_collection =
+ SmoothnessEstimator::Legendre::default_quadrature_collection(
+ Laplace<dim>::fe);
+ legendre.initialize(n_coefficients_per_direction,
+ Laplace<dim>::fe,
+ expansion_q_collection);
}
template <int dim>
-std::pair<unsigned int, unsigned int>
-Problem4<dim>::substitute_h_for_p(
- std::vector<typename Triangulation<dim>::active_cell_iterator> &p_cells)
+void
+Problem4<dim>::substitute_h_for_p()
{
Vector<float> smoothness_indicators(
Laplace<dim>::triangulation.n_active_cells());
- SmoothnessEstimator::legendre_coefficient_decay(Laplace<dim>::dof_handler,
- Laplace<dim>::solution,
- smoothness_indicators);
-
- unsigned int num_p_cells = 0;
- unsigned int num_h_cells = 0;
- for (auto &cell : Laplace<dim>::dof_handler.active_cell_iterators())
- if (cell->refine_flag_set())
- {
- typename Triangulation<dim>::active_cell_iterator tria_cell(
- &(Laplace<dim>::triangulation), cell->level(), cell->index());
-
- const unsigned int cur_fe_index = cell->active_fe_index();
- const bool p_ref = smoothness_indicators(cell->index()) < exp(-1.);
-
- if (cur_fe_index < Laplace<dim>::fe.size() - 1 && p_ref)
- {
- ++num_p_cells;
- cell->clear_refine_flag();
- p_cells.push_back(tria_cell);
- }
- else
- {
- ++num_h_cells;
- }
- }
-
- return std::make_pair(num_h_cells, num_p_cells);
+ SmoothnessEstimator::Legendre::coefficient_decay_per_direction(
+ legendre,
+ Laplace<dim>::dof_handler,
+ Laplace<dim>::solution,
+ smoothness_indicators);
+
+ hp::Refinement::p_adaptivity_from_absolute_threshold(
+ Laplace<dim>::dof_handler,
+ smoothness_indicators,
+ static_cast<float>(1.),
+ static_cast<float>(0.),
+ std::greater<float>(),
+ std::less<float>());
+ hp::Refinement::choose_p_over_h(Laplace<dim>::dof_handler);
}
--- /dev/null
+
+DEAL::
+DEAL::0 256 0 0 289 0.267081 1.78414 1.23386 1.29971
+DEAL::1 256 0 56 477 0.0236257 0.743531 0.249786 1.18708
+DEAL::2 268 4 47 790 0.00108376 0.293919 0.0466411 0.361176
+DEAL::3 328 20 33 1498 0.000182183 0.0621768 0.00717343 0.0760236
+DEAL::4 508 60 24 3402 2.40462e-05 0.0134610 0.00111509 0.0224341
+DEAL::5 571 21 80 4536 3.48748e-06 0.00249342 0.000176335 0.00398940
+DEAL::6 595 8 106 5758 7.30475e-07 0.000644499 3.23861e-05 0.000860213
+DEAL::7 673 26 93 7725 3.32389e-07 0.000304984 2.38742e-05 0.000356132
+DEAL::8 721 16 118 9246 5.75345e-08 7.11276e-05 2.90017e-06 0.000111393
+DEAL::9 772 17 127 11000 3.50959e-08 3.79011e-05 2.91724e-06 5.96652e-05
+DEAL::10 796 8 146 13074 9.97255e-09 1.29806e-05 7.46491e-07 1.99550e-05
#include <deal.II/dofs/dof_tools.h>
#include <deal.II/fe/fe_q.h>
+#include <deal.II/fe/fe_series.h>
#include <deal.II/grid/grid_generator.h>
#include <deal.II/grid/grid_refinement.h>
#include <deal.II/hp/dof_handler.h>
#include <deal.II/hp/fe_values.h>
+#include <deal.II/hp/q_collection.h>
+#include <deal.II/hp/refinement.h>
#include <deal.II/lac/affine_constraints.h>
#include <deal.II/lac/dynamic_sparsity_pattern.h>
Vector<double> solution;
Vector<double> system_rhs;
- const unsigned int max_degree;
+ const unsigned int max_degree;
+ hp::QCollection<dim> fourier_q_collection;
+ FESeries::Fourier<dim> fourier;
};
quadrature_collection.push_back(QGauss<dim>(degree + 1));
face_quadrature_collection.push_back(QGauss<dim - 1>(degree + 1));
}
+
+ QGauss<1> base_quadrature(2);
+ QIterated<dim> quadrature(base_quadrature, max_degree);
+ for (unsigned int i = 0; i < fe_collection.size(); ++i)
+ fourier_q_collection.push_back(quadrature);
+
+ const std::vector<unsigned int> n_coefficients_per_direction(
+ fe_collection.size(), max_degree);
+ fourier.initialize(n_coefficients_per_direction,
+ fe_collection,
+ fourier_q_collection);
}
solution,
estimated_error_per_cell);
+ GridRefinement::refine_and_coarsen_fixed_number(triangulation,
+ estimated_error_per_cell,
+ 0.3,
+ 0.03);
+
Vector<float> smoothness_indicators;
- SmoothnessEstimator::fourier_coefficient_decay(dof_handler,
- solution,
- smoothness_indicators);
+ SmoothnessEstimator::Fourier::coefficient_decay(
+ fourier,
+ dof_handler,
+ solution,
+ smoothness_indicators,
+ /*regression_strategy=*/VectorTools::Linfty_norm,
+ /*smallest_abs_coefficient=*/1e-10,
+ /*only_flagged_cells=*/true);
+
+ hp::Refinement::p_adaptivity_from_relative_threshold(dof_handler,
+ smoothness_indicators,
+ 0.5,
+ 0);
+ hp::Refinement::choose_p_over_h(dof_handler);
// Output to VTK
if (false)
data_out.write_vtk(output);
}
- {
- GridRefinement::refine_and_coarsen_fixed_number(triangulation,
- estimated_error_per_cell,
- 0.3,
- 0.03);
-
- float max_smoothness = *std::min_element(smoothness_indicators.begin(),
- smoothness_indicators.end()),
- min_smoothness = *std::max_element(smoothness_indicators.begin(),
- smoothness_indicators.end());
- {
- typename hp::DoFHandler<dim>::active_cell_iterator
- cell = dof_handler.begin_active(),
- endc = dof_handler.end();
- for (; cell != endc; ++cell)
- if (cell->refine_flag_set())
- {
- max_smoothness =
- std::max(max_smoothness,
- smoothness_indicators(cell->active_cell_index()));
- min_smoothness =
- std::min(min_smoothness,
- smoothness_indicators(cell->active_cell_index()));
- }
- }
- const float threshold_smoothness = (max_smoothness + min_smoothness) / 2;
-
- {
- typename hp::DoFHandler<dim>::active_cell_iterator
- cell = dof_handler.begin_active(),
- endc = dof_handler.end();
- for (; cell != endc; ++cell)
- if (cell->refine_flag_set() &&
- (smoothness_indicators(cell->active_cell_index()) >
- threshold_smoothness) &&
- (cell->active_fe_index() + 1 < fe_collection.size()))
- {
- cell->clear_refine_flag();
- cell->set_active_fe_index(cell->active_fe_index() + 1);
- }
- }
-
- triangulation.execute_coarsening_and_refinement();
- }
+ triangulation.execute_coarsening_and_refinement();
}
#include <deal.II/hp/dof_handler.h>
#include <deal.II/hp/fe_values.h>
+#include <deal.II/hp/q_collection.h>
#include <deal.II/hp/refinement.h>
#include <deal.II/lac/affine_constraints.h>
#include <deal.II/numerics/data_out.h>
#include <deal.II/numerics/error_estimator.h>
#include <deal.II/numerics/matrix_tools.h>
+#include <deal.II/numerics/smoothness_estimator.h>
#include <deal.II/numerics/vector_tools.h>
#include <complex>
hp::QCollection<dim> quadrature_collection;
hp::QCollection<dim - 1> face_quadrature_collection;
- hp::QCollection<dim> fourier_q_collection;
- std::shared_ptr<FESeries::Fourier<dim>> fourier;
- std::vector<double> ln_k;
- Table<dim, std::complex<double>> fourier_coefficients;
+ hp::QCollection<dim> fourier_q_collection;
+ FESeries::Fourier<dim> fourier;
AffineConstraints<double> constraints;
face_quadrature_collection.push_back(QGauss<dim - 1>(degree + 1));
}
- const unsigned int N = max_degree;
-
QGauss<1> base_quadrature(2);
- QIterated<dim> quadrature(base_quadrature, N);
+ QIterated<dim> quadrature(base_quadrature, max_degree);
for (unsigned int i = 0; i < fe_collection.size(); i++)
fourier_q_collection.push_back(quadrature);
- fourier = std::make_shared<FESeries::Fourier<dim>>(N,
- fe_collection,
- fourier_q_collection);
-
- resize(fourier_coefficients, N);
+ const std::vector<unsigned int> n_coefficients_per_direction(
+ fe_collection.size(), max_degree);
+ fourier.initialize(n_coefficients_per_direction,
+ fe_collection,
+ fourier_q_collection);
}
solution,
estimated_error_per_cell);
- Vector<float> smoothness_indicators(triangulation.n_active_cells());
- estimate_smoothness(smoothness_indicators);
-
parallel::distributed::GridRefinement::refine_and_coarsen_fixed_number(
triangulation, estimated_error_per_cell, 0.3, 0.03);
+ Vector<float> smoothness_indicators(triangulation.n_active_cells());
+ SmoothnessEstimator::Fourier::coefficient_decay(
+ fourier,
+ dof_handler,
+ solution,
+ smoothness_indicators,
+ /*regression_strategy=*/VectorTools::Linfty_norm,
+ /*smallest_abs_coefficient=*/1e-10,
+ /*only_flagged_cells=*/true);
+
hp::Refinement::p_adaptivity_from_relative_threshold(dof_handler,
smoothness_indicators,
0.5,
postprocess();
}
}
-
-
-
- template <int dim>
- std::pair<bool, unsigned int>
- LaplaceProblem<dim>::predicate(const TableIndices<dim> &ind)
- {
- unsigned int v = 0;
- for (unsigned int i = 0; i < dim; i++)
- v += ind[i] * ind[i];
- if (v > 0 && v < max_degree * max_degree)
- return std::make_pair(true, v);
- else
- return std::make_pair(false, v);
- }
-
-
-
- template <int dim>
- void
- LaplaceProblem<dim>::estimate_smoothness(Vector<float> &smoothness_indicators)
- {
- Vector<double> local_dof_values;
-
- for (const auto &cell : dof_handler.active_cell_iterators())
- if (cell->is_locally_owned())
- {
- local_dof_values.reinit(cell->get_fe().dofs_per_cell);
- cell->get_dof_values(solution, local_dof_values);
-
- fourier->calculate(local_dof_values,
- cell->active_fe_index(),
- fourier_coefficients);
-
- std::pair<std::vector<unsigned int>, std::vector<double>> res =
- FESeries::process_coefficients<dim>(
- fourier_coefficients,
- std::bind(&LaplaceProblem<dim>::predicate,
- this,
- std::placeholders::_1),
- VectorTools::Linfty_norm);
-
- Assert(res.first.size() == res.second.size(), ExcInternalError());
-
- if (ln_k.size() == 0)
- {
- ln_k.resize(res.first.size(), 0);
- for (unsigned int f = 0; f < ln_k.size(); f++)
- ln_k[f] =
- std::log(2.0 * numbers::PI * std::sqrt(1. * res.first[f]));
- }
-
- for (double &residual_element : res.second)
- residual_element = std::log(residual_element);
-
- std::pair<double, double> fit =
- FESeries::linear_regression(ln_k, res.second);
-
- smoothness_indicators(cell->active_cell_index()) =
- -fit.first - 1. * dim / 2;
- }
- }
} // namespace Step27
#include <deal.II/hp/dof_handler.h>
#include <deal.II/hp/fe_values.h>
+#include <deal.II/hp/q_collection.h>
#include <deal.II/hp/refinement.h>
#include <deal.II/lac/affine_constraints.h>
#include <deal.II/numerics/data_out.h>
#include <deal.II/numerics/error_estimator.h>
#include <deal.II/numerics/matrix_tools.h>
+#include <deal.II/numerics/smoothness_estimator.h>
#include <deal.II/numerics/vector_tools.h>
#include <complex>
hp::QCollection<dim> quadrature_collection;
hp::QCollection<dim - 1> face_quadrature_collection;
- hp::QCollection<dim> fourier_q_collection;
- std::shared_ptr<FESeries::Fourier<dim>> fourier;
- std::vector<double> ln_k;
- Table<dim, std::complex<double>> fourier_coefficients;
+ hp::QCollection<dim> fourier_q_collection;
+ FESeries::Fourier<dim> fourier;
AffineConstraints<double> constraints;
face_quadrature_collection.push_back(QGauss<dim - 1>(degree + 1));
}
- const unsigned int N = max_degree;
-
QGauss<1> base_quadrature(2);
- QIterated<dim> quadrature(base_quadrature, N);
+ QIterated<dim> quadrature(base_quadrature, max_degree);
for (unsigned int i = 0; i < fe_collection.size(); i++)
fourier_q_collection.push_back(quadrature);
- fourier = std::make_shared<FESeries::Fourier<dim>>(N,
- fe_collection,
- fourier_q_collection);
-
- resize(fourier_coefficients, N);
+ const std::vector<unsigned int> n_coefficients_per_direction(
+ fe_collection.size(), max_degree);
+ fourier.initialize(n_coefficients_per_direction,
+ fe_collection,
+ fourier_q_collection);
}
solution,
estimated_error_per_cell);
-
- Vector<float> smoothness_indicators(triangulation.n_active_cells());
- estimate_smoothness(smoothness_indicators);
-
parallel::distributed::GridRefinement::refine_and_coarsen_fixed_number(
triangulation, estimated_error_per_cell, 0.3, 0.03);
+ Vector<float> smoothness_indicators(triangulation.n_active_cells());
+ SmoothnessEstimator::Fourier::coefficient_decay(
+ fourier,
+ dof_handler,
+ solution,
+ smoothness_indicators,
+ /*regression_strategy=*/VectorTools::Linfty_norm,
+ /*smallest_abs_coefficient=*/1e-10,
+ /*only_flagged_cells=*/true);
+
hp::Refinement::p_adaptivity_from_relative_threshold(dof_handler,
smoothness_indicators,
0.5,
postprocess();
}
}
-
-
-
- template <int dim>
- std::pair<bool, unsigned int>
- LaplaceProblem<dim>::predicate(const TableIndices<dim> &ind)
- {
- unsigned int v = 0;
- for (unsigned int i = 0; i < dim; i++)
- v += ind[i] * ind[i];
- if (v > 0 && v < max_degree * max_degree)
- return std::make_pair(true, v);
- else
- return std::make_pair(false, v);
- }
-
-
-
- template <int dim>
- void
- LaplaceProblem<dim>::estimate_smoothness(Vector<float> &smoothness_indicators)
- {
- Vector<double> local_dof_values;
-
- for (const auto &cell : dof_handler.active_cell_iterators())
- if (cell->is_locally_owned())
- {
- local_dof_values.reinit(cell->get_fe().dofs_per_cell);
- cell->get_dof_values(solution, local_dof_values);
-
- fourier->calculate(local_dof_values,
- cell->active_fe_index(),
- fourier_coefficients);
-
- std::pair<std::vector<unsigned int>, std::vector<double>> res =
- FESeries::process_coefficients<dim>(
- fourier_coefficients,
- std::bind(&LaplaceProblem<dim>::predicate,
- this,
- std::placeholders::_1),
- VectorTools::Linfty_norm);
-
- Assert(res.first.size() == res.second.size(), ExcInternalError());
-
- if (ln_k.size() == 0)
- {
- ln_k.resize(res.first.size(), 0);
- for (unsigned int f = 0; f < ln_k.size(); f++)
- ln_k[f] =
- std::log(2.0 * numbers::PI * std::sqrt(1. * res.first[f]));
- }
-
- for (double &residual_element : res.second)
- residual_element = std::log(residual_element);
-
- std::pair<double, double> fit =
- FESeries::linear_regression(ln_k, res.second);
-
- smoothness_indicators(cell->active_cell_index()) =
- -fit.first - 1. * dim / 2;
- }
- }
} // namespace Step27
// ---------------------------------------------------------------------
//
-// Copyright (C) 2018 by the deal.II authors
+// Copyright (C) 2018 - 2020 by the deal.II authors
//
// This file is part of the deal.II library.
//
//
// ---------------------------------------------------------------------
+
+
// essentially similar to fe/fe_series_05.cc but test smoothness estimation.
+
+
#include <deal.II/base/function.h>
#include <deal.II/base/quadrature_lib.h>
+#include <deal.II/fe/component_mask.h>
#include <deal.II/fe/fe_q.h>
#include <deal.II/fe/fe_series.h>
#include "../tests.h"
+
using namespace dealii;
+
template <int dim>
class LegendreFunction : public Function<dim>
{
const Table<dim, double> coefficients;
};
+
+
// copy-paste from fe_series.cc
template <int dim>
double
return res;
}
+
+
template <>
double
LegendreFunction<2>::value(const dealii::Point<2> &point,
return f;
}
+
+
void
compare(const Table<2, double> &coeff1, const Table<2, double> &coeff2)
{
<< linf << std::endl;
}
+
+
void resize(Table<2, double> &coeff, const unsigned int N)
{
coeff.reinit(N, N);
void
test(const LegendreFunction<dim> &func, const unsigned int poly_degree)
{
- // custom predicate:
- // p-ref for linear elements and use j=1,...,pe otherwise.
- const auto coefficients_predicate = [](std::vector<bool> &flags) -> void {
- std::fill(flags.begin(), flags.end(), flags.size() > 2);
- flags[0] = false;
- };
-
const unsigned int max_poly = poly_degree + 3;
deallog << "-----------------------------------" << std::endl;
deallog << dim << "d, p=" << poly_degree << ", max_p=" << max_poly
<< std::endl;
deallog << "-----------------------------------" << std::endl;
- Triangulation<dim> triangulation;
- hp::DoFHandler<dim> dof_handler(triangulation);
- hp::FECollection<dim> fe_collection;
- hp::QCollection<dim> quadrature_formula;
// add some extra FEs in fe_collection
+ hp::FECollection<dim> fe_collection;
for (unsigned int p = 1; p <= max_poly; p++)
- {
- fe_collection.push_back(FE_Q<dim>(p));
- quadrature_formula.push_back(QGauss<dim>(p + 1 + 5));
- }
+ fe_collection.push_back(FE_Q<dim>(p));
+
+ const unsigned int fe_index = poly_degree - 1;
+ const std::vector<unsigned int> n_coefficients_per_direction =
+ SmoothnessEstimator::Legendre::default_number_of_coefficients_per_direction(
+ fe_collection);
+ const unsigned int n_modes = n_coefficients_per_direction[fe_index];
+
+ // custom predicate:
+ // p-ref for linear elements and use j=1,...,pe otherwise.
+ ComponentMask coefficients_predicate(n_modes, true);
+ coefficients_predicate.set(0, false);
+ Triangulation<dim> triangulation;
GridGenerator::hyper_cube(triangulation, 0.0, 1.0); // reference cell
- const unsigned int fe_index = poly_degree - 1;
+
+ hp::DoFHandler<dim> dof_handler(triangulation);
+ dof_handler.set_fe(fe_collection);
dof_handler.begin_active()->set_active_fe_index(fe_index);
dof_handler.distribute_dofs(fe_collection);
Vector<double> values(dof_handler.n_dofs());
-
VectorTools::interpolate(dof_handler, func, values);
- const unsigned int N = poly_degree + 1;
- FESeries::Legendre<dim> legendre(N, fe_collection, quadrature_formula);
+ hp::QCollection<dim> q_collection =
+ SmoothnessEstimator::Legendre::default_quadrature_collection(fe_collection);
+ FESeries::Legendre<dim> legendre(n_coefficients_per_direction,
+ fe_collection,
+ q_collection);
const Table<dim, double> &coeff_in = func.get_coefficients();
Table<dim, double> coeff_out;
- resize(coeff_out, N);
+ resize(coeff_out, n_modes);
Vector<double> local_dof_values;
// finally test smoothness estimator:
Vector<float> smoothness(1);
- SmoothnessEstimator::legendre_coefficient_decay(
- legendre, dof_handler, values, smoothness, coefficients_predicate);
+ SmoothnessEstimator::Legendre::coefficient_decay_per_direction(
+ legendre,
+ dof_handler,
+ values,
+ smoothness,
+ coefficients_predicate,
+ /*smallest_abs_coefficient=*/1e-10,
+ /*only_flagged_cells=*/false);
deallog << "smoothness:" << std::endl << smoothness[0] << std::endl;
dof_handler.clear();
}
+
+
int
main()
{
LegendreFunction<dim> function(coeff_in);
test(function, p);
- deallog << "expected smoothness:" << std::endl << 0. << std::endl;
+ deallog << "expected smoothness:" << std::endl
+ << std::numeric_limits<float>::infinity() << std::endl;
}
// for quadratic we can already assign exponential decay: a_i = C exp ( -k
test(function, p);
deallog << "expected smoothness:" << std::endl
- << exp(-std::min(k1, k2)) << std::endl;
+ << std::min(k1, k2) << std::endl;
}
// linear elements in 3D (expect zero output)
LegendreFunction<dim> function(coeff_in);
test(function, p);
- deallog << "expected smoothness:" << std::endl << 0. << std::endl;
+ deallog << "expected smoothness:" << std::endl
+ << std::numeric_limits<float>::infinity() << std::endl;
}
// cubic in 3D
test(function, p);
deallog << "expected smoothness:" << std::endl
- << exp(-std::min(k1, std::min(k2, k3))) << std::endl;
+ << std::min(k1, std::min(k2, k3)) << std::endl;
}
LegendreFunction<dim> function(coeff_in);
test(function, p);
- deallog << "expected smoothness:" << std::endl << exp(-k1) << std::endl;
+ deallog << "expected smoothness:" << std::endl << k1 << std::endl;
}
// cubic in 3D (zero)
LegendreFunction<dim> function(coeff_in);
test(function, p);
- deallog << "expected smoothness:" << std::endl << 0. << std::endl;
+ deallog << "expected smoothness:" << std::endl
+ << std::numeric_limits<float>::infinity() << std::endl;
}
dealii::deallog << "Ok" << std::endl;
DEAL::2d, p=1, max_p=4
DEAL::-----------------------------------
DEAL::Linf norm in exact and calculate Legendre coefficients:
-DEAL::1.77636e-15
+DEAL::1.33227e-15
DEAL::smoothness:
-DEAL::0.00000
+DEAL::inf
DEAL::expected smoothness:
-DEAL::0.00000
+DEAL::inf
DEAL::-----------------------------------
DEAL::2d, p=2, max_p=5
DEAL::-----------------------------------
DEAL::Linf norm in exact and calculate Legendre coefficients:
-DEAL::7.21645e-16
+DEAL::7.27596e-12
DEAL::smoothness:
-DEAL::0.367879
+DEAL::1.00000
DEAL::expected smoothness:
-DEAL::0.367879
+DEAL::1.00000
DEAL::-----------------------------------
DEAL::3d, p=1, max_p=4
DEAL::-----------------------------------
DEAL::Linf norm in exact and calculate Legendre coefficients:
-DEAL::3.55271e-15
+DEAL::1.77636e-15
DEAL::smoothness:
-DEAL::0.00000
+DEAL::inf
DEAL::expected smoothness:
-DEAL::0.00000
+DEAL::inf
DEAL::-----------------------------------
DEAL::3d, p=3, max_p=6
DEAL::-----------------------------------
DEAL::Linf norm in exact and calculate Legendre coefficients:
-DEAL::3.19189e-15
+DEAL::1.38017e-11
DEAL::smoothness:
-DEAL::0.135335
+DEAL::2.00000
DEAL::expected smoothness:
-DEAL::0.135335
+DEAL::2.00000
DEAL::-----------------------------------
DEAL::3d, p=4, max_p=7
DEAL::-----------------------------------
DEAL::Linf norm in exact and calculate Legendre coefficients:
-DEAL::4.31599e-15
+DEAL::1.31877e-11
DEAL::smoothness:
-DEAL::0.135335
+DEAL::2.00000
DEAL::expected smoothness:
-DEAL::0.135335
+DEAL::2.00000
DEAL::-----------------------------------
DEAL::3d, p=3, max_p=6
DEAL::-----------------------------------
DEAL::Linf norm in exact and calculate Legendre coefficients:
DEAL::0.00000
DEAL::smoothness:
-DEAL::0.00000
+DEAL::inf
DEAL::expected smoothness:
-DEAL::0.00000
+DEAL::inf
DEAL::Ok
--- /dev/null
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2019 - 2020 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.
+//
+// ---------------------------------------------------------------------
+
+
+
+// Check Fourier coefficient for simple functions in 1D/2D/3D.
+// Further, estimate regularity in 1D/2D/3D.
+// Similar to tests fe/fe_series_01.cc and numerics/smoothness_estimator_01.cc.
+
+// Test functions: 1D: x^p, 2D: (x*y)^p, 3D: (x*y*z)^p
+// Below is the MWE in Maxima for x*y*z:
+/*********************************************************
+integrate2(F,xx,aa,bb,yy,cc,dd):=block(integrate(integrate(F,xx,aa,bb),yy,cc,dd));
+integrate3(F,xx,aa,bb,yy,cc,dd,zz,ee,ff):=block(integrate(integrate(integrate(F,xx,aa,bb),yy,cc,dd),zz,ee,ff));
+a:0;
+b:1;
+nmax:3;
+Phi(xx,nn):=exp(((-2)*%i*%pi*nn*xx)/(b-a));
+Phi2(xx,nn,yy,mm):=Phi(xx,nn)*Phi(yy,mm);
+Phi3(xx,nn,yy,mm,zz,ll):=Phi(xx,nn)*Phi(yy,mm)*Phi(zz,ll);
+f3:x*y*z;
+C3(n,m,l):=integrate3(f3*conjugate(Phi3(x,n,y,m,z,l)),x,a,b,y,a,b,z,a,b)/(b-a)^3;
+load(functs);
+for i:0 thru nmax do (for j:0 thru nmax do (for k:0 thru nmax do
+print([i,j,k],fullratsimp(C3(i,j,k)))));
+*********************************************************/
+
+
+#include <deal.II/base/function.h>
+#include <deal.II/base/quadrature_lib.h>
+#include <deal.II/base/table.h>
+
+#include <deal.II/fe/fe_q.h>
+#include <deal.II/fe/fe_series.h>
+
+#include <deal.II/grid/grid_generator.h>
+#include <deal.II/grid/tria.h>
+
+#include <deal.II/hp/dof_handler.h>
+#include <deal.II/hp/q_collection.h>
+
+#include <deal.II/lac/vector.h>
+
+#include <deal.II/numerics/smoothness_estimator.h>
+#include <deal.II/numerics/vector_tools.h>
+
+#include <cmath>
+#include <iostream>
+
+#include "../tests.h"
+
+
+using namespace dealii;
+
+
+template <int dim>
+class PolyFunction : public Function<dim>
+{
+public:
+ PolyFunction(const unsigned int poly_degree)
+ : Function<dim>(1)
+ , poly_degree(poly_degree)
+ {}
+
+ virtual double
+ value(const Point<dim> &point, const unsigned int component = 0) const;
+
+private:
+ const unsigned int poly_degree;
+};
+
+
+
+template <int dim>
+double
+PolyFunction<dim>::value(const Point<dim> &point, const unsigned int) const
+{
+ double f = 1.;
+ for (unsigned int d = 0; d < dim; ++d)
+ f *= std::pow(point[d], poly_degree);
+ return f;
+}
+
+
+
+template <typename CoefficientType>
+void
+ prepare_symmetric_coefficients(Table<1, CoefficientType> & coeff,
+ const std::vector<CoefficientType> &coeff_1d)
+{
+ Assert(coeff.size(0) == coeff_1d.size(), ExcInternalError());
+
+ for (unsigned int i = 0; i < coeff.size(0); ++i)
+ coeff(i) = coeff_1d[i];
+}
+
+template <typename CoefficientType>
+void
+ prepare_symmetric_coefficients(Table<2, CoefficientType> & coeff,
+ const std::vector<CoefficientType> &coeff_1d)
+{
+ for (unsigned int d = 0; d < 2; ++d)
+ Assert(coeff.size(d) == coeff_1d.size(), ExcInternalError());
+
+ for (unsigned int i = 0; i < coeff.size(0); ++i)
+ for (unsigned int j = 0; j < coeff.size(1); ++j)
+ coeff(i, j) = coeff_1d[i] * coeff_1d[j];
+}
+
+template <typename CoefficientType>
+void
+ prepare_symmetric_coefficients(Table<3, CoefficientType> & coeff,
+ const std::vector<CoefficientType> &coeff_1d)
+{
+ for (unsigned int d = 0; d < 3; ++d)
+ Assert(coeff.size(d) == coeff_1d.size(), ExcInternalError());
+
+ for (unsigned int i = 0; i < coeff.size(0); ++i)
+ for (unsigned int j = 0; j < coeff.size(1); ++j)
+ for (unsigned int k = 0; k < coeff.size(2); ++k)
+ coeff(i, j, k) = coeff_1d[i] * coeff_1d[j] * coeff_1d[k];
+}
+
+
+
+template <typename CoefficientType>
+typename CoefficientType::value_type
+compare(const Table<1, CoefficientType> &coeff1,
+ const Table<1, CoefficientType> &coeff2)
+{
+ Assert(coeff1.size(0) == coeff2.size(0), ExcInternalError());
+
+ typename CoefficientType::value_type linf = 0.;
+ for (unsigned int i = 0; i < coeff1.size(0); i++)
+ linf = std::max(linf, std::abs(coeff1(i) - coeff2(i)));
+
+ return linf;
+}
+
+template <typename CoefficientType>
+typename CoefficientType::value_type
+compare(const Table<2, CoefficientType> &coeff1,
+ const Table<2, CoefficientType> &coeff2)
+{
+ for (unsigned int d = 0; d < 2; ++d)
+ Assert(coeff1.size(d) == coeff2.size(d), ExcInternalError());
+
+ typename CoefficientType::value_type linf = 0.;
+ for (unsigned int i = 0; i < coeff1.size(0); i++)
+ for (unsigned int j = 0; j < coeff1.size(1); j++)
+ linf = std::max(linf, std::abs(coeff1(i, j) - coeff2(i, j)));
+
+ return linf;
+}
+
+template <typename CoefficientType>
+typename CoefficientType::value_type
+compare(const Table<3, CoefficientType> &coeff1,
+ const Table<3, CoefficientType> &coeff2)
+{
+ for (unsigned int d = 0; d < 3; ++d)
+ Assert(coeff1.size(d) == coeff2.size(d), ExcInternalError());
+
+ typename CoefficientType::value_type linf = 0.;
+ for (unsigned int i = 0; i < coeff1.size(0); i++)
+ for (unsigned int j = 0; j < coeff1.size(1); j++)
+ for (unsigned int k = 0; k < coeff1.size(2); k++)
+ linf = std::max(linf, std::abs(coeff1(i, j, k) - coeff2(i, j, k)));
+
+ return linf;
+}
+
+
+
+template <int dim>
+void
+test(const unsigned int poly_degree)
+{
+ const unsigned int max_poly = 3;
+ hp::FECollection<dim> fe_collection;
+ for (unsigned int p = 1; p <= max_poly; ++p)
+ fe_collection.push_back(FE_Q<dim>(p));
+
+ const unsigned int fe_index = poly_degree - 1;
+ const std::vector<unsigned int> n_coefficients_per_direction =
+ SmoothnessEstimator::Fourier::default_number_of_coefficients_per_direction(
+ fe_collection);
+ const unsigned int n_modes = n_coefficients_per_direction[fe_index];
+
+ Assert((poly_degree >= 1) && (poly_degree <= max_poly), ExcInternalError());
+ Assert((n_modes >= 3) && (n_modes <= max_poly + 1), ExcInternalError());
+
+ deallog << "-----------------------------------" << std::endl;
+ deallog << dim << "d, p=" << poly_degree << ", max_p=" << max_poly
+ << ", n_modes=" << n_modes << std::endl;
+ deallog << "-----------------------------------" << std::endl;
+
+ // --- prepare test function ---
+ PolyFunction<dim> test_function(poly_degree);
+
+ // exact coefficients in 1D case
+ const double &pi = numbers::PI;
+ const double pi2 = std::pow(pi, 2);
+ const double pi3 = std::pow(pi, 3);
+
+ std::vector<std::complex<double>> exact(n_modes);
+ switch (poly_degree)
+ {
+ case 1:
+ exact[0] = std::complex<double>(1., 0.) / 2.;
+ if (n_modes > 1)
+ exact[1] = std::complex<double>(0., -1.) / (2. * pi);
+ if (n_modes > 2)
+ exact[2] = std::complex<double>(0., -1.) / (4. * pi);
+ if (n_modes > 3)
+ exact[3] = std::complex<double>(0., -1.) / (6. * pi);
+ break;
+ case 2:
+ exact[0] = std::complex<double>(1., 0.) / 3.;
+ if (n_modes > 1)
+ exact[1] = std::complex<double>(1., -pi) / (2. * pi2);
+ if (n_modes > 2)
+ exact[2] = std::complex<double>(1., -2. * pi) / (8. * pi2);
+ if (n_modes > 3)
+ exact[3] = std::complex<double>(1., -3. * pi) / (18. * pi2);
+ break;
+ case 3:
+ exact[0] = std::complex<double>(1., 0.) / 4.;
+ if (n_modes > 1)
+ exact[1] = std::complex<double>(3. * pi, 3. - 2. * pi2) / (4. * pi3);
+ if (n_modes > 2)
+ exact[2] = std::complex<double>(6. * pi, 3. - 8. * pi2) / (32. * pi3);
+ if (n_modes > 3)
+ exact[3] = std::complex<double>(3. * pi, 1. - 6. * pi2) / (36. * pi3);
+ break;
+ default:
+ Assert(false, ExcNotImplemented());
+ break;
+ }
+
+ // coefficient table for multi-dimensional case
+ TableIndices<dim> size;
+ for (unsigned int d = 0; d < dim; ++d)
+ size[d] = n_modes;
+ Table<dim, std::complex<double>> coeff_in;
+ coeff_in.reinit(size);
+ prepare_symmetric_coefficients(coeff_in, exact);
+
+
+ // --- prepare data structures ---
+ Triangulation<dim> tria;
+ GridGenerator::hyper_cube(tria, 0.0, 1.0); // reference cell
+
+ hp::DoFHandler<dim> dof_handler(tria);
+ dof_handler.begin_active()->set_active_fe_index(fe_index);
+ dof_handler.distribute_dofs(fe_collection);
+
+
+ // --- calculate coefficients from exact solution ---
+ Vector<double> values(dof_handler.n_dofs());
+ VectorTools::interpolate(dof_handler, test_function, values);
+
+ Vector<double> local_dof_values;
+ auto cell = dof_handler.begin_active();
+ local_dof_values.reinit(cell->get_fe().dofs_per_cell);
+ cell->get_dof_values(values, local_dof_values);
+
+ hp::QCollection<dim> q_collection =
+ SmoothnessEstimator::Fourier::default_quadrature_collection(fe_collection);
+ FESeries::Fourier<dim> fourier(n_coefficients_per_direction,
+ fe_collection,
+ q_collection);
+
+ Table<dim, std::complex<double>> coeff_out;
+ coeff_out.reinit(size);
+ fourier.calculate(local_dof_values, cell->active_fe_index(), coeff_out);
+
+ // verify results
+ const double linf = compare(coeff_in, coeff_out);
+ deallog << "Linf norm in exact and calculate Fourier coefficients:"
+ << std::endl
+ << linf << std::endl;
+
+ // finally test smoothness estimator:
+ Vector<float> regularity(1);
+ SmoothnessEstimator::Fourier::coefficient_decay(
+ fourier,
+ dof_handler,
+ values,
+ regularity,
+ /*regression_strategy=*/VectorTools::Linfty_norm,
+ /*smallest_abs_coefficient=*/1e-10,
+ /*only_flagged_cells=*/false);
+
+ deallog << "estimated regularity:" << std::endl << regularity[0] << std::endl;
+
+ dof_handler.clear();
+}
+
+
+
+int
+main()
+{
+ std::ofstream logfile("output");
+ dealii::deallog.attach(logfile, /*do not print job id*/ false);
+ dealii::deallog.depth_console(0);
+
+ for (unsigned int poly_degree = 1; poly_degree <= 3; ++poly_degree)
+ {
+ test<1>(poly_degree);
+ test<2>(poly_degree);
+ test<3>(poly_degree);
+ }
+
+ dealii::deallog << "Ok" << std::endl;
+}
--- /dev/null
+DEAL::-----------------------------------
+DEAL::1d, p=1, max_p=3, n_modes=3
+DEAL::-----------------------------------
+DEAL::Linf norm in exact and calculate Fourier coefficients:
+DEAL::0.000831516
+DEAL::estimated regularity:
+DEAL::0.988446
+DEAL::-----------------------------------
+DEAL::2d, p=1, max_p=3, n_modes=3
+DEAL::-----------------------------------
+DEAL::Linf norm in exact and calculate Fourier coefficients:
+DEAL::0.000415758
+DEAL::estimated regularity:
+DEAL::1.09966
+DEAL::-----------------------------------
+DEAL::3d, p=1, max_p=3, n_modes=3
+DEAL::-----------------------------------
+DEAL::Linf norm in exact and calculate Fourier coefficients:
+DEAL::0.000207879
+DEAL::estimated regularity:
+DEAL::0.890459
+DEAL::-----------------------------------
+DEAL::1d, p=2, max_p=3, n_modes=3
+DEAL::-----------------------------------
+DEAL::Linf norm in exact and calculate Fourier coefficients:
+DEAL::0.000637547
+DEAL::estimated regularity:
+DEAL::1.04031
+DEAL::-----------------------------------
+DEAL::2d, p=2, max_p=3, n_modes=3
+DEAL::-----------------------------------
+DEAL::Linf norm in exact and calculate Fourier coefficients:
+DEAL::0.000212516
+DEAL::estimated regularity:
+DEAL::0.854283
+DEAL::-----------------------------------
+DEAL::3d, p=2, max_p=3, n_modes=3
+DEAL::-----------------------------------
+DEAL::Linf norm in exact and calculate Fourier coefficients:
+DEAL::7.08386e-05
+DEAL::estimated regularity:
+DEAL::0.504717
+DEAL::-----------------------------------
+DEAL::1d, p=3, max_p=3, n_modes=4
+DEAL::-----------------------------------
+DEAL::Linf norm in exact and calculate Fourier coefficients:
+DEAL::0.000375709
+DEAL::estimated regularity:
+DEAL::0.972651
+DEAL::-----------------------------------
+DEAL::2d, p=3, max_p=3, n_modes=4
+DEAL::-----------------------------------
+DEAL::Linf norm in exact and calculate Fourier coefficients:
+DEAL::9.39272e-05
+DEAL::estimated regularity:
+DEAL::0.522683
+DEAL::-----------------------------------
+DEAL::3d, p=3, max_p=3, n_modes=4
+DEAL::-----------------------------------
+DEAL::Linf norm in exact and calculate Fourier coefficients:
+DEAL::2.34818e-05
+DEAL::estimated regularity:
+DEAL::0.417735
+DEAL::Ok
test()
{
// setup
- hp::FECollection<dim> hp_fe;
- hp::QCollection<dim> hp_q;
+ std::vector<unsigned int> n_modes;
+ hp::FECollection<dim> hp_fe;
+ hp::QCollection<dim> hp_q;
const unsigned int min_degree = 1, max_degree = 2;
const QGauss<1> base_quadrature(4);
const QSorted<dim> quadrature_sorted(quadrature);
for (unsigned int p = min_degree; p <= max_degree; ++p)
{
+ n_modes.push_back(max_degree + 1);
hp_fe.push_back(FE_Q<dim>(p));
hp_q.push_back(quadrature_sorted);
}
- FESeries::Fourier<dim> fourier_save(max_degree + 1, hp_fe, hp_q);
- FESeries::Fourier<dim> fourier_load(max_degree + 1, hp_fe, hp_q);
+ FESeries::Fourier<dim> fourier_save(n_modes, hp_fe, hp_q);
+ FESeries::Fourier<dim> fourier_load(n_modes, hp_fe, hp_q);
// create transformation matrices
fourier_save.precalculate_all_transformation_matrices();
-DEAL:1d::3 2 10 FE_Q<1>(1) 10 FE_Q<1>(2) 2 0 0 0 0 0 0 8 0 0 0 0 0 1 0 0 3.47159221014868569e-02 1 4.65284077898513171e-01 1 5.34715922101486885e-01 1 9.65284077898513226e-01 1 1.65004739103785936e-01 1 3.34995260896214064e-01 1 6.65004739103785880e-01 1 8.34995260896214120e-01 8 0 8.69637112843634624e-02 8.69637112843634624e-02 8.69637112843634624e-02 8.69637112843634624e-02 1.63036288715636524e-01 1.63036288715636524e-01 1.63036288715636524e-01 1.63036288715636524e-01 8 0 1 3.47159221014868569e-02 1 4.65284077898513171e-01 1 5.34715922101486885e-01 1 9.65284077898513226e-01 1 1.65004739103785936e-01 1 3.34995260896214064e-01 1 6.65004739103785880e-01 1 8.34995260896214120e-01 8 0 8.69637112843634624e-02 8.69637112843634624e-02 8.69637112843634624e-02 8.69637112843634624e-02 1.63036288715636524e-01 1.63036288715636524e-01 1.63036288715636524e-01 1.63036288715636524e-01 0 0 2 0 0 0 0 0 6 4.99999999999999944e-01 0.00000000000000000e+00 5.00000000000000000e-01 0.00000000000000000e+00 -1.73472347597680709e-18 1.59153688032438484e-01 0.00000000000000000e+00 -1.59153688032438317e-01 5.33917649563659877e-04 8.02149275159986103e-02 5.33917649563897534e-04 -8.02149275159983605e-02 0 0 2 3 2 9 1.66666666666666657e-01 0.00000000000000000e+00 1.66666666666666713e-01 0.00000000000000000e+00 6.66666666666666630e-01 0.00000000000000000e+00 1.01327477769070529e-01 1.59153688032438401e-01 1.01327477769070570e-01 -1.59153688032438401e-01 -2.02654955538141113e-01 1.38777878078144568e-16 2.48179557444021051e-02 8.02149275159984715e-02 2.48179557444023757e-02 -8.02149275159984576e-02 -4.85680761896770119e-02 2.22044604925031308e-16 2 3 3
+DEAL:1d::2 0 3 3 2 10 FE_Q<1>(1) 10 FE_Q<1>(2) 2 0 0 0 0 0 0 8 0 0 0 0 0 1 0 0 3.47159221014868569e-02 1 4.65284077898513171e-01 1 5.34715922101486885e-01 1 9.65284077898513226e-01 1 1.65004739103785936e-01 1 3.34995260896214064e-01 1 6.65004739103785880e-01 1 8.34995260896214120e-01 8 0 8.69637112843634624e-02 8.69637112843634624e-02 8.69637112843634624e-02 8.69637112843634624e-02 1.63036288715636524e-01 1.63036288715636524e-01 1.63036288715636524e-01 1.63036288715636524e-01 8 0 1 3.47159221014868569e-02 1 4.65284077898513171e-01 1 5.34715922101486885e-01 1 9.65284077898513226e-01 1 1.65004739103785936e-01 1 3.34995260896214064e-01 1 6.65004739103785880e-01 1 8.34995260896214120e-01 8 0 8.69637112843634624e-02 8.69637112843634624e-02 8.69637112843634624e-02 8.69637112843634624e-02 1.63036288715636524e-01 1.63036288715636524e-01 1.63036288715636524e-01 1.63036288715636524e-01 0 0 2 0 0 0 0 0 6 4.99999999999999944e-01 0.00000000000000000e+00 5.00000000000000000e-01 0.00000000000000000e+00 -1.73472347597680709e-18 1.59153688032438484e-01 0.00000000000000000e+00 -1.59153688032438317e-01 5.33917649563659877e-04 8.02149275159986103e-02 5.33917649563897534e-04 -8.02149275159983605e-02 0 0 2 3 2 9 1.66666666666666657e-01 0.00000000000000000e+00 1.66666666666666713e-01 0.00000000000000000e+00 6.66666666666666630e-01 0.00000000000000000e+00 1.01327477769070529e-01 1.59153688032438401e-01 1.01327477769070570e-01 -1.59153688032438401e-01 -2.02654955538141113e-01 1.38777878078144568e-16 2.48179557444021051e-02 8.02149275159984715e-02 2.48179557444023757e-02 -8.02149275159984576e-02 -4.85680761896770119e-02 2.22044604925031308e-16 2 3 3
DEAL:1d::OK
-DEAL:2d::9 2 10 FE_Q<2>(1) 10 FE_Q<2>(2) 2 0 0 0 0 0 0 64 0 0 0 0 0 2 0 0 3.47159221014868569e-02 5.34715922101486885e-01 2 9.65284077898513226e-01 9.65284077898513226e-01 2 5.34715922101486885e-01 9.65284077898513226e-01 2 4.65284077898513171e-01 9.65284077898513226e-01 2 3.47159221014868569e-02 9.65284077898513226e-01 2 9.65284077898513226e-01 5.34715922101486885e-01 2 5.34715922101486885e-01 5.34715922101486885e-01 2 4.65284077898513171e-01 5.34715922101486885e-01 2 3.47159221014868569e-02 3.47159221014868569e-02 2 9.65284077898513226e-01 4.65284077898513171e-01 2 5.34715922101486885e-01 4.65284077898513171e-01 2 4.65284077898513171e-01 4.65284077898513171e-01 2 3.47159221014868569e-02 4.65284077898513171e-01 2 9.65284077898513226e-01 3.47159221014868569e-02 2 4.65284077898513171e-01 3.47159221014868569e-02 2 5.34715922101486885e-01 3.47159221014868569e-02 2 5.34715922101486885e-01 8.34995260896214120e-01 2 8.34995260896214120e-01 3.47159221014868569e-02 2 6.65004739103785880e-01 5.34715922101486885e-01 2 6.65004739103785880e-01 3.47159221014868569e-02 2 3.47159221014868569e-02 6.65004739103785880e-01 2 4.65284077898513171e-01 6.65004739103785880e-01 2 5.34715922101486885e-01 6.65004739103785880e-01 2 9.65284077898513226e-01 6.65004739103785880e-01 2 3.47159221014868569e-02 8.34995260896214120e-01 2 4.65284077898513171e-01 8.34995260896214120e-01 2 8.34995260896214120e-01 5.34715922101486885e-01 2 9.65284077898513226e-01 8.34995260896214120e-01 2 1.65004739103785936e-01 9.65284077898513226e-01 2 3.34995260896214064e-01 9.65284077898513226e-01 2 3.34995260896214064e-01 3.47159221014868569e-02 2 6.65004739103785880e-01 9.65284077898513226e-01 2 8.34995260896214120e-01 9.65284077898513226e-01 2 1.65004739103785936e-01 3.47159221014868569e-02 2 1.65004739103785936e-01 5.34715922101486885e-01 2 9.65284077898513226e-01 3.34995260896214064e-01 2 5.34715922101486885e-01 1.65004739103785936e-01 2 1.65004739103785936e-01 4.65284077898513171e-01 2 3.34995260896214064e-01 4.65284077898513171e-01 2 5.34715922101486885e-01 3.34995260896214064e-01 2 4.65284077898513171e-01 3.34995260896214064e-01 2 4.65284077898513171e-01 1.65004739103785936e-01 2 3.47159221014868569e-02 3.34995260896214064e-01 2 6.65004739103785880e-01 4.65284077898513171e-01 2 8.34995260896214120e-01 4.65284077898513171e-01 2 9.65284077898513226e-01 1.65004739103785936e-01 2 3.34995260896214064e-01 5.34715922101486885e-01 2 3.47159221014868569e-02 1.65004739103785936e-01 2 1.65004739103785936e-01 3.34995260896214064e-01 2 3.34995260896214064e-01 3.34995260896214064e-01 2 6.65004739103785880e-01 3.34995260896214064e-01 2 8.34995260896214120e-01 3.34995260896214064e-01 2 8.34995260896214120e-01 1.65004739103785936e-01 2 8.34995260896214120e-01 8.34995260896214120e-01 2 6.65004739103785880e-01 8.34995260896214120e-01 2 3.34995260896214064e-01 8.34995260896214120e-01 2 1.65004739103785936e-01 8.34995260896214120e-01 2 8.34995260896214120e-01 6.65004739103785880e-01 2 6.65004739103785880e-01 6.65004739103785880e-01 2 3.34995260896214064e-01 1.65004739103785936e-01 2 3.34995260896214064e-01 6.65004739103785880e-01 2 1.65004739103785936e-01 6.65004739103785880e-01 2 6.65004739103785880e-01 1.65004739103785936e-01 2 1.65004739103785936e-01 1.65004739103785936e-01 64 0 7.56268708035012522e-03 7.56268708035012522e-03 7.56268708035012522e-03 7.56268708035012522e-03 7.56268708035012522e-03 7.56268708035012522e-03 7.56268708035012522e-03 7.56268708035012522e-03 7.56268708035012522e-03 7.56268708035012522e-03 7.56268708035012522e-03 7.56268708035012522e-03 7.56268708035012522e-03 7.56268708035012522e-03 7.56268708035012522e-03 7.56268708035012522e-03 1.41782407407407395e-02 1.41782407407407395e-02 1.41782407407407395e-02 1.41782407407407395e-02 1.41782407407407395e-02 1.41782407407407395e-02 1.41782407407407395e-02 1.41782407407407395e-02 1.41782407407407395e-02 1.41782407407407395e-02 1.41782407407407395e-02 1.41782407407407395e-02 1.41782407407407395e-02 1.41782407407407395e-02 1.41782407407407395e-02 1.41782407407407395e-02 1.41782407407407395e-02 1.41782407407407395e-02 1.41782407407407395e-02 1.41782407407407395e-02 1.41782407407407395e-02 1.41782407407407395e-02 1.41782407407407395e-02 1.41782407407407395e-02 1.41782407407407395e-02 1.41782407407407395e-02 1.41782407407407395e-02 1.41782407407407395e-02 1.41782407407407395e-02 1.41782407407407395e-02 1.41782407407407395e-02 1.41782407407407395e-02 2.65808314381683897e-02 2.65808314381683897e-02 2.65808314381683897e-02 2.65808314381683897e-02 2.65808314381683897e-02 2.65808314381683897e-02 2.65808314381683897e-02 2.65808314381683897e-02 2.65808314381683897e-02 2.65808314381683897e-02 2.65808314381683897e-02 2.65808314381683897e-02 2.65808314381683897e-02 2.65808314381683897e-02 2.65808314381683897e-02 2.65808314381683897e-02 64 0 2 3.47159221014868569e-02 5.34715922101486885e-01 2 9.65284077898513226e-01 9.65284077898513226e-01 2 5.34715922101486885e-01 9.65284077898513226e-01 2 4.65284077898513171e-01 9.65284077898513226e-01 2 3.47159221014868569e-02 9.65284077898513226e-01 2 9.65284077898513226e-01 5.34715922101486885e-01 2 5.34715922101486885e-01 5.34715922101486885e-01 2 4.65284077898513171e-01 5.34715922101486885e-01 2 3.47159221014868569e-02 3.47159221014868569e-02 2 9.65284077898513226e-01 4.65284077898513171e-01 2 5.34715922101486885e-01 4.65284077898513171e-01 2 4.65284077898513171e-01 4.65284077898513171e-01 2 3.47159221014868569e-02 4.65284077898513171e-01 2 9.65284077898513226e-01 3.47159221014868569e-02 2 4.65284077898513171e-01 3.47159221014868569e-02 2 5.34715922101486885e-01 3.47159221014868569e-02 2 5.34715922101486885e-01 8.34995260896214120e-01 2 8.34995260896214120e-01 3.47159221014868569e-02 2 6.65004739103785880e-01 5.34715922101486885e-01 2 6.65004739103785880e-01 3.47159221014868569e-02 2 3.47159221014868569e-02 6.65004739103785880e-01 2 4.65284077898513171e-01 6.65004739103785880e-01 2 5.34715922101486885e-01 6.65004739103785880e-01 2 9.65284077898513226e-01 6.65004739103785880e-01 2 3.47159221014868569e-02 8.34995260896214120e-01 2 4.65284077898513171e-01 8.34995260896214120e-01 2 8.34995260896214120e-01 5.34715922101486885e-01 2 9.65284077898513226e-01 8.34995260896214120e-01 2 1.65004739103785936e-01 9.65284077898513226e-01 2 3.34995260896214064e-01 9.65284077898513226e-01 2 3.34995260896214064e-01 3.47159221014868569e-02 2 6.65004739103785880e-01 9.65284077898513226e-01 2 8.34995260896214120e-01 9.65284077898513226e-01 2 1.65004739103785936e-01 3.47159221014868569e-02 2 1.65004739103785936e-01 5.34715922101486885e-01 2 9.65284077898513226e-01 3.34995260896214064e-01 2 5.34715922101486885e-01 1.65004739103785936e-01 2 1.65004739103785936e-01 4.65284077898513171e-01 2 3.34995260896214064e-01 4.65284077898513171e-01 2 5.34715922101486885e-01 3.34995260896214064e-01 2 4.65284077898513171e-01 3.34995260896214064e-01 2 4.65284077898513171e-01 1.65004739103785936e-01 2 3.47159221014868569e-02 3.34995260896214064e-01 2 6.65004739103785880e-01 4.65284077898513171e-01 2 8.34995260896214120e-01 4.65284077898513171e-01 2 9.65284077898513226e-01 1.65004739103785936e-01 2 3.34995260896214064e-01 5.34715922101486885e-01 2 3.47159221014868569e-02 1.65004739103785936e-01 2 1.65004739103785936e-01 3.34995260896214064e-01 2 3.34995260896214064e-01 3.34995260896214064e-01 2 6.65004739103785880e-01 3.34995260896214064e-01 2 8.34995260896214120e-01 3.34995260896214064e-01 2 8.34995260896214120e-01 1.65004739103785936e-01 2 8.34995260896214120e-01 8.34995260896214120e-01 2 6.65004739103785880e-01 8.34995260896214120e-01 2 3.34995260896214064e-01 8.34995260896214120e-01 2 1.65004739103785936e-01 8.34995260896214120e-01 2 8.34995260896214120e-01 6.65004739103785880e-01 2 6.65004739103785880e-01 6.65004739103785880e-01 2 3.34995260896214064e-01 1.65004739103785936e-01 2 3.34995260896214064e-01 6.65004739103785880e-01 2 1.65004739103785936e-01 6.65004739103785880e-01 2 6.65004739103785880e-01 1.65004739103785936e-01 2 1.65004739103785936e-01 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DEAL:2d::OK
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-9.03021825067809591e-18 2.82593529509107713e-04 1.93375788693043814e-04 2.82593529509098606e-04 -1.93375788693050970e-04 -3.42422689921490701e-04 -7.00326840789855520e-18 -3.42422689921491406e-04 -6.94101148627536413e-18 2.82593529509100232e-04 1.93375788693044329e-04 5.85420340659764627e-05 1.89215625477152721e-04 5.85420340659924005e-05 -1.89215625477162479e-04 5.85420340659733186e-05 1.89215625477151745e-04 5.85420340659912621e-05 -1.89215625477163021e-04 5.85420340659595492e-05 1.89215625477162018e-04 5.85420340660127022e-05 -1.89215625477167141e-04 -1.14565196267454395e-04 8.86335276006899875e-18 2 27 27
DEAL:3d::OK
test()
{
// setup
- hp::FECollection<dim> hp_fe;
- hp::QCollection<dim> hp_q;
+ std::vector<unsigned int> n_modes;
+ hp::FECollection<dim> hp_fe;
+ hp::QCollection<dim> hp_q;
const unsigned int min_degree = 1, max_degree = 2;
const QGauss<dim> quadrature(max_degree + 1);
const QSorted<dim> quadrature_sorted(quadrature);
for (unsigned int p = min_degree; p <= max_degree; ++p)
{
+ n_modes.push_back(max_degree + 1);
hp_fe.push_back(FE_Q<dim>(p));
hp_q.push_back(quadrature_sorted);
}
- FESeries::Legendre<dim> legendre_save(max_degree + 1, hp_fe, hp_q);
- FESeries::Legendre<dim> legendre_load(max_degree + 1, hp_fe, hp_q);
+ FESeries::Legendre<dim> legendre_save(n_modes, hp_fe, hp_q);
+ FESeries::Legendre<dim> legendre_load(n_modes, hp_fe, hp_q);
// create transformation matrices
legendre_save.precalculate_all_transformation_matrices();
-DEAL:1d::3 2 10 FE_Q<1>(1) 10 FE_Q<1>(2) 2 0 0 0 0 0 0 3 0 0 0 0 0 1 0 0 1.12701665379258312e-01 1 8.87298334620741702e-01 1 5.00000000000000000e-01 3 0 2.77777777777777790e-01 2.77777777777777790e-01 4.44444444444444420e-01 3 0 1 1.12701665379258312e-01 1 8.87298334620741702e-01 1 5.00000000000000000e-01 3 0 2.77777777777777790e-01 2.77777777777777790e-01 4.44444444444444420e-01 0 0 2 0 0 0 0 0 6 3.53553390593273786e-01 3.53553390593273786e-01 -3.53553390593273786e-01 3.53553390593273786e-01 1.38777878078144568e-16 1.38777878078144568e-16 0 0 2 3 2 9 1.17851130197757933e-01 1.17851130197757933e-01 4.71404520791031678e-01 -3.53553390593273842e-01 3.53553390593273842e-01 -2.08166817117216851e-17 2.35702260395515950e-01 2.35702260395515950e-01 -4.71404520791031567e-01 2 3 3
+DEAL:1d::2 0 3 3 2 10 FE_Q<1>(1) 10 FE_Q<1>(2) 2 0 0 0 0 0 0 3 0 0 0 0 0 1 0 0 1.12701665379258312e-01 1 8.87298334620741702e-01 1 5.00000000000000000e-01 3 0 2.77777777777777790e-01 2.77777777777777790e-01 4.44444444444444420e-01 3 0 1 1.12701665379258312e-01 1 8.87298334620741702e-01 1 5.00000000000000000e-01 3 0 2.77777777777777790e-01 2.77777777777777790e-01 4.44444444444444420e-01 0 0 2 0 0 0 0 0 6 3.53553390593273786e-01 3.53553390593273786e-01 -3.53553390593273786e-01 3.53553390593273786e-01 1.38777878078144568e-16 1.38777878078144568e-16 0 0 2 3 2 9 1.17851130197757933e-01 1.17851130197757933e-01 4.71404520791031678e-01 -3.53553390593273842e-01 3.53553390593273842e-01 -2.08166817117216851e-17 2.35702260395515950e-01 2.35702260395515950e-01 -4.71404520791031567e-01 2 3 3
DEAL:1d::OK
-DEAL:2d::3 2 10 FE_Q<2>(1) 10 FE_Q<2>(2) 2 0 0 0 0 0 0 9 0 0 0 0 0 2 0 0 1.12701665379258312e-01 1.12701665379258312e-01 2 8.87298334620741702e-01 1.12701665379258312e-01 2 1.12701665379258312e-01 8.87298334620741702e-01 2 8.87298334620741702e-01 8.87298334620741702e-01 2 5.00000000000000000e-01 1.12701665379258312e-01 2 1.12701665379258312e-01 5.00000000000000000e-01 2 8.87298334620741702e-01 5.00000000000000000e-01 2 5.00000000000000000e-01 8.87298334620741702e-01 2 5.00000000000000000e-01 5.00000000000000000e-01 9 0 7.71604938271605034e-02 7.71604938271605034e-02 7.71604938271605034e-02 7.71604938271605034e-02 1.23456790123456783e-01 1.23456790123456783e-01 1.23456790123456783e-01 1.23456790123456783e-01 1.97530864197530853e-01 9 0 2 1.12701665379258312e-01 1.12701665379258312e-01 2 8.87298334620741702e-01 1.12701665379258312e-01 2 1.12701665379258312e-01 8.87298334620741702e-01 2 8.87298334620741702e-01 8.87298334620741702e-01 2 5.00000000000000000e-01 1.12701665379258312e-01 2 1.12701665379258312e-01 5.00000000000000000e-01 2 8.87298334620741702e-01 5.00000000000000000e-01 2 5.00000000000000000e-01 8.87298334620741702e-01 2 5.00000000000000000e-01 5.00000000000000000e-01 9 0 7.71604938271605034e-02 7.71604938271605034e-02 7.71604938271605034e-02 7.71604938271605034e-02 1.23456790123456783e-01 1.23456790123456783e-01 1.23456790123456783e-01 1.23456790123456783e-01 1.97530864197530853e-01 0 0 2 0 0 0 0 0 36 1.25000000000000028e-01 1.25000000000000028e-01 1.25000000000000028e-01 1.25000000000000028e-01 -1.25000000000000028e-01 -1.25000000000000028e-01 1.25000000000000000e-01 1.25000000000000028e-01 0.00000000000000000e+00 0.00000000000000000e+00 8.67361737988403547e-18 8.67361737988403547e-18 -1.25000000000000028e-01 1.25000000000000028e-01 -1.25000000000000028e-01 1.25000000000000028e-01 1.25000000000000083e-01 -1.25000000000000083e-01 -1.25000000000000083e-01 1.25000000000000056e-01 -2.92734586571086197e-17 2.60208521396521064e-17 -2.92734586571086197e-17 2.60208521396521064e-17 0.00000000000000000e+00 0.00000000000000000e+00 0.00000000000000000e+00 1.73472347597680709e-17 -2.92734586571086197e-17 -2.92734586571086197e-17 2.60208521396521064e-17 2.60208521396521064e-17 8.67361737988403547e-17 8.67361737988403547e-17 8.67361737988403547e-17 6.50521303491302660e-17 0 0 2 9 4 81 1.38888888888888968e-02 1.38888888888888968e-02 1.38888888888888968e-02 1.38888888888888951e-02 5.55555555555555802e-02 5.55555555555555802e-02 5.55555555555555802e-02 5.55555555555555733e-02 2.22222222222222265e-01 -4.16666666666666921e-02 -4.16666666666666921e-02 4.16666666666666852e-02 4.16666666666666852e-02 0.00000000000000000e+00 0.00000000000000000e+00 -1.66666666666666685e-01 1.66666666666666685e-01 -1.04083408558608426e-17 2.77777777777777901e-02 2.77777777777777901e-02 2.77777777777777901e-02 2.77777777777777901e-02 -5.55555555555555663e-02 -5.55555555555555663e-02 1.11111111111111147e-01 1.11111111111111147e-01 -2.22222222222222265e-01 -4.16666666666666990e-02 4.16666666666666990e-02 -4.16666666666666852e-02 4.16666666666666921e-02 -1.66666666666666685e-01 1.66666666666666685e-01 0.00000000000000000e+00 0.00000000000000000e+00 0.00000000000000000e+00 1.25000000000000056e-01 -1.25000000000000056e-01 -1.25000000000000083e-01 1.25000000000000083e-01 2.92734586571086197e-18 0.00000000000000000e+00 0.00000000000000000e+00 0.00000000000000000e+00 7.80625564189563192e-18 -8.33333333333333842e-02 8.33333333333333842e-02 -8.33333333333333842e-02 8.33333333333333842e-02 1.66666666666666657e-01 -1.66666666666666657e-01 0.00000000000000000e+00 0.00000000000000000e+00 0.00000000000000000e+00 2.77777777777777901e-02 2.77777777777777901e-02 2.77777777777777901e-02 2.77777777777777901e-02 1.11111111111111147e-01 1.11111111111111147e-01 -5.55555555555555663e-02 -5.55555555555555663e-02 -2.22222222222222265e-01 -8.33333333333333842e-02 -8.33333333333333842e-02 8.33333333333333842e-02 8.33333333333333842e-02 8.13151629364128326e-19 0.00000000000000000e+00 1.66666666666666657e-01 -1.66666666666666657e-01 0.00000000000000000e+00 5.55555555555555802e-02 5.55555555555555802e-02 5.55555555555555872e-02 5.55555555555555872e-02 -1.11111111111111133e-01 -1.11111111111111133e-01 -1.11111111111111133e-01 -1.11111111111111133e-01 2.22222222222222349e-01 2 9 9
+DEAL:2d::2 0 3 3 2 10 FE_Q<2>(1) 10 FE_Q<2>(2) 2 0 0 0 0 0 0 9 0 0 0 0 0 2 0 0 1.12701665379258312e-01 1.12701665379258312e-01 2 8.87298334620741702e-01 1.12701665379258312e-01 2 1.12701665379258312e-01 8.87298334620741702e-01 2 8.87298334620741702e-01 8.87298334620741702e-01 2 5.00000000000000000e-01 1.12701665379258312e-01 2 1.12701665379258312e-01 5.00000000000000000e-01 2 8.87298334620741702e-01 5.00000000000000000e-01 2 5.00000000000000000e-01 8.87298334620741702e-01 2 5.00000000000000000e-01 5.00000000000000000e-01 9 0 7.71604938271605034e-02 7.71604938271605034e-02 7.71604938271605034e-02 7.71604938271605034e-02 1.23456790123456783e-01 1.23456790123456783e-01 1.23456790123456783e-01 1.23456790123456783e-01 1.97530864197530853e-01 9 0 2 1.12701665379258312e-01 1.12701665379258312e-01 2 8.87298334620741702e-01 1.12701665379258312e-01 2 1.12701665379258312e-01 8.87298334620741702e-01 2 8.87298334620741702e-01 8.87298334620741702e-01 2 5.00000000000000000e-01 1.12701665379258312e-01 2 1.12701665379258312e-01 5.00000000000000000e-01 2 8.87298334620741702e-01 5.00000000000000000e-01 2 5.00000000000000000e-01 8.87298334620741702e-01 2 5.00000000000000000e-01 5.00000000000000000e-01 9 0 7.71604938271605034e-02 7.71604938271605034e-02 7.71604938271605034e-02 7.71604938271605034e-02 1.23456790123456783e-01 1.23456790123456783e-01 1.23456790123456783e-01 1.23456790123456783e-01 1.97530864197530853e-01 0 0 2 0 0 0 0 0 36 1.25000000000000028e-01 1.25000000000000028e-01 1.25000000000000028e-01 1.25000000000000028e-01 -1.25000000000000028e-01 -1.25000000000000028e-01 1.25000000000000000e-01 1.25000000000000028e-01 0.00000000000000000e+00 0.00000000000000000e+00 8.67361737988403547e-18 8.67361737988403547e-18 -1.25000000000000028e-01 1.25000000000000028e-01 -1.25000000000000028e-01 1.25000000000000028e-01 1.25000000000000083e-01 -1.25000000000000083e-01 -1.25000000000000083e-01 1.25000000000000056e-01 -2.92734586571086197e-17 2.60208521396521064e-17 -2.92734586571086197e-17 2.60208521396521064e-17 0.00000000000000000e+00 0.00000000000000000e+00 0.00000000000000000e+00 1.73472347597680709e-17 -2.92734586571086197e-17 -2.92734586571086197e-17 2.60208521396521064e-17 2.60208521396521064e-17 8.67361737988403547e-17 8.67361737988403547e-17 8.67361737988403547e-17 6.50521303491302660e-17 0 0 2 9 4 81 1.38888888888888968e-02 1.38888888888888968e-02 1.38888888888888968e-02 1.38888888888888951e-02 5.55555555555555802e-02 5.55555555555555802e-02 5.55555555555555802e-02 5.55555555555555733e-02 2.22222222222222265e-01 -4.16666666666666921e-02 -4.16666666666666921e-02 4.16666666666666852e-02 4.16666666666666852e-02 0.00000000000000000e+00 0.00000000000000000e+00 -1.66666666666666685e-01 1.66666666666666685e-01 -1.04083408558608426e-17 2.77777777777777901e-02 2.77777777777777901e-02 2.77777777777777901e-02 2.77777777777777901e-02 -5.55555555555555663e-02 -5.55555555555555663e-02 1.11111111111111147e-01 1.11111111111111147e-01 -2.22222222222222265e-01 -4.16666666666666990e-02 4.16666666666666990e-02 -4.16666666666666852e-02 4.16666666666666921e-02 -1.66666666666666685e-01 1.66666666666666685e-01 0.00000000000000000e+00 0.00000000000000000e+00 0.00000000000000000e+00 1.25000000000000056e-01 -1.25000000000000056e-01 -1.25000000000000083e-01 1.25000000000000083e-01 2.92734586571086197e-18 0.00000000000000000e+00 0.00000000000000000e+00 0.00000000000000000e+00 7.80625564189563192e-18 -8.33333333333333842e-02 8.33333333333333842e-02 -8.33333333333333842e-02 8.33333333333333842e-02 1.66666666666666657e-01 -1.66666666666666657e-01 0.00000000000000000e+00 0.00000000000000000e+00 0.00000000000000000e+00 2.77777777777777901e-02 2.77777777777777901e-02 2.77777777777777901e-02 2.77777777777777901e-02 1.11111111111111147e-01 1.11111111111111147e-01 -5.55555555555555663e-02 -5.55555555555555663e-02 -2.22222222222222265e-01 -8.33333333333333842e-02 -8.33333333333333842e-02 8.33333333333333842e-02 8.33333333333333842e-02 8.13151629364128326e-19 0.00000000000000000e+00 1.66666666666666657e-01 -1.66666666666666657e-01 0.00000000000000000e+00 5.55555555555555802e-02 5.55555555555555802e-02 5.55555555555555872e-02 5.55555555555555872e-02 -1.11111111111111133e-01 -1.11111111111111133e-01 -1.11111111111111133e-01 -1.11111111111111133e-01 2.22222222222222349e-01 2 9 9
DEAL:2d::OK
-DEAL:3d::3 2 10 FE_Q<3>(1) 10 FE_Q<3>(2) 2 0 0 0 0 0 0 27 0 0 0 0 0 3 0 0 1.12701665379258312e-01 1.12701665379258312e-01 1.12701665379258312e-01 3 8.87298334620741702e-01 1.12701665379258312e-01 1.12701665379258312e-01 3 8.87298334620741702e-01 8.87298334620741702e-01 8.87298334620741702e-01 3 1.12701665379258312e-01 8.87298334620741702e-01 8.87298334620741702e-01 3 1.12701665379258312e-01 8.87298334620741702e-01 1.12701665379258312e-01 3 8.87298334620741702e-01 8.87298334620741702e-01 1.12701665379258312e-01 3 8.87298334620741702e-01 1.12701665379258312e-01 8.87298334620741702e-01 3 1.12701665379258312e-01 1.12701665379258312e-01 8.87298334620741702e-01 3 5.00000000000000000e-01 1.12701665379258312e-01 1.12701665379258312e-01 3 5.00000000000000000e-01 8.87298334620741702e-01 8.87298334620741702e-01 3 8.87298334620741702e-01 5.00000000000000000e-01 8.87298334620741702e-01 3 1.12701665379258312e-01 5.00000000000000000e-01 8.87298334620741702e-01 3 5.00000000000000000e-01 1.12701665379258312e-01 8.87298334620741702e-01 3 8.87298334620741702e-01 8.87298334620741702e-01 5.00000000000000000e-01 3 1.12701665379258312e-01 8.87298334620741702e-01 5.00000000000000000e-01 3 8.87298334620741702e-01 1.12701665379258312e-01 5.00000000000000000e-01 3 1.12701665379258312e-01 1.12701665379258312e-01 5.00000000000000000e-01 3 5.00000000000000000e-01 8.87298334620741702e-01 1.12701665379258312e-01 3 8.87298334620741702e-01 5.00000000000000000e-01 1.12701665379258312e-01 3 1.12701665379258312e-01 5.00000000000000000e-01 1.12701665379258312e-01 3 8.87298334620741702e-01 5.00000000000000000e-01 5.00000000000000000e-01 3 5.00000000000000000e-01 8.87298334620741702e-01 5.00000000000000000e-01 3 1.12701665379258312e-01 5.00000000000000000e-01 5.00000000000000000e-01 3 5.00000000000000000e-01 1.12701665379258312e-01 5.00000000000000000e-01 3 5.00000000000000000e-01 5.00000000000000000e-01 8.87298334620741702e-01 3 5.00000000000000000e-01 5.00000000000000000e-01 1.12701665379258312e-01 3 5.00000000000000000e-01 5.00000000000000000e-01 5.00000000000000000e-01 27 0 2.14334705075445858e-02 2.14334705075445858e-02 2.14334705075445858e-02 2.14334705075445858e-02 2.14334705075445858e-02 2.14334705075445858e-02 2.14334705075445858e-02 2.14334705075445858e-02 3.42935528120713318e-02 3.42935528120713318e-02 3.42935528120713318e-02 3.42935528120713318e-02 3.42935528120713318e-02 3.42935528120713318e-02 3.42935528120713318e-02 3.42935528120713318e-02 3.42935528120713318e-02 3.42935528120713318e-02 3.42935528120713318e-02 3.42935528120713318e-02 5.48696844993141197e-02 5.48696844993141197e-02 5.48696844993141197e-02 5.48696844993141197e-02 5.48696844993141267e-02 5.48696844993141267e-02 8.77914951989025999e-02 27 0 3 1.12701665379258312e-01 1.12701665379258312e-01 1.12701665379258312e-01 3 8.87298334620741702e-01 1.12701665379258312e-01 1.12701665379258312e-01 3 8.87298334620741702e-01 8.87298334620741702e-01 8.87298334620741702e-01 3 1.12701665379258312e-01 8.87298334620741702e-01 8.87298334620741702e-01 3 1.12701665379258312e-01 8.87298334620741702e-01 1.12701665379258312e-01 3 8.87298334620741702e-01 8.87298334620741702e-01 1.12701665379258312e-01 3 8.87298334620741702e-01 1.12701665379258312e-01 8.87298334620741702e-01 3 1.12701665379258312e-01 1.12701665379258312e-01 8.87298334620741702e-01 3 5.00000000000000000e-01 1.12701665379258312e-01 1.12701665379258312e-01 3 5.00000000000000000e-01 8.87298334620741702e-01 8.87298334620741702e-01 3 8.87298334620741702e-01 5.00000000000000000e-01 8.87298334620741702e-01 3 1.12701665379258312e-01 5.00000000000000000e-01 8.87298334620741702e-01 3 5.00000000000000000e-01 1.12701665379258312e-01 8.87298334620741702e-01 3 8.87298334620741702e-01 8.87298334620741702e-01 5.00000000000000000e-01 3 1.12701665379258312e-01 8.87298334620741702e-01 5.00000000000000000e-01 3 8.87298334620741702e-01 1.12701665379258312e-01 5.00000000000000000e-01 3 1.12701665379258312e-01 1.12701665379258312e-01 5.00000000000000000e-01 3 5.00000000000000000e-01 8.87298334620741702e-01 1.12701665379258312e-01 3 8.87298334620741702e-01 5.00000000000000000e-01 1.12701665379258312e-01 3 1.12701665379258312e-01 5.00000000000000000e-01 1.12701665379258312e-01 3 8.87298334620741702e-01 5.00000000000000000e-01 5.00000000000000000e-01 3 5.00000000000000000e-01 8.87298334620741702e-01 5.00000000000000000e-01 3 1.12701665379258312e-01 5.00000000000000000e-01 5.00000000000000000e-01 3 5.00000000000000000e-01 1.12701665379258312e-01 5.00000000000000000e-01 3 5.00000000000000000e-01 5.00000000000000000e-01 8.87298334620741702e-01 3 5.00000000000000000e-01 5.00000000000000000e-01 1.12701665379258312e-01 3 5.00000000000000000e-01 5.00000000000000000e-01 5.00000000000000000e-01 27 0 2.14334705075445858e-02 2.14334705075445858e-02 2.14334705075445858e-02 2.14334705075445858e-02 2.14334705075445858e-02 2.14334705075445858e-02 2.14334705075445858e-02 2.14334705075445858e-02 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DEAL:3d::OK
projects / dealii.git / commitdiff