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Reference documentation for deal.II version 9.2.0
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Reference documentation for deal.II version 9.2.0
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Functions | |
| 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) |
| 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) |
| template<int dim, int spacedim> | |
| FESeries::Legendre< dim, spacedim > | default_fe_series (const hp::FECollection< dim, spacedim > &fe_collection) |
Smoothness estimation strategy based on the decay of Legendre expansion coefficients.
In one dimension, the finite element solution on cell \(K\) with polynomial degree \(p\) can be written as
\begin{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 \end{eqnarray*}
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 the mapping from the reference cell to the actual cell is affine. 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 [21])
\[ |a_k| \sim c \, \exp(-\sigma k) \]
We determine their decay rate \(\sigma\) by performing the linear regression fit of
\[ \ln |a_k| \sim C - \sigma k \]
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 [50]).
| void SmoothnessEstimator::Legendre::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, |
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| const double | smallest_abs_coefficient = 1e-10, |
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| const bool | only_flagged_cells = false |
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| ) |
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
\begin{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 \end{eqnarray*}
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 solution, this function writes the decay rate for every cell into the output vector smoothness_indicators.
| [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. |
| [in] | dof_handler | An hp::DoFHandler. |
| [in] | solution | A solution vector. |
| [out] | smoothness_indicators | A vector for smoothness indicators. |
| [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. |
| [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 ignore 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 \(\sigma=\infty\). |
| [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 [50] [37] [21].
Definition at line 100 of file smoothness_estimator.cc.
| void SmoothnessEstimator::Legendre::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(), |
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| const double | smallest_abs_coefficient = 1e-10, |
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| const bool | only_flagged_cells = false |
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| ) |
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 solution, this function writes the decay rate for every cell into the output vector smoothness_indicators.
| [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. |
| [in] | dof_handler | An hp::DoFHandler |
| [in] | solution | A solution vector |
| [out] | smoothness_indicators | A vector for smoothness indicators |
| [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. |
| [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 ignore 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 \(\sigma=\infty\). |
| [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 [19].
Definition at line 183 of file smoothness_estimator.cc.
| FESeries::Legendre< dim, spacedim > SmoothnessEstimator::Legendre::default_fe_series | ( | const hp::FECollection< dim, spacedim > & | fe_collection | ) |
Returns a FESeries::Legendre object for Legendre series expansions with the default configuration for smoothness estimation purposes.
For each finite element of the provided fe_collection, we use as many modes as its polynomial degree plus one, since we start with the first Legendre polynomial which is just a constant. Further for each element, we use a Gaussian quadrature designed to yield exact results for the highest order Legendre polynomial used.
Definition at line 290 of file smoothness_estimator.cc.
1.8.17