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Reference documentation for deal.II version 9.1.1
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Local integrators related to the Laplacian and its DG formulations. More...
Functions | |
template<int dim> | |
void | cell_matrix (FullMatrix< double > &M, const FEValuesBase< dim > &fe, const double factor=1.) |
template<int dim> | |
void | cell_residual (Vector< double > &result, const FEValuesBase< dim > &fe, const std::vector< Tensor< 1, dim >> &input, double factor=1.) |
template<int dim> | |
void | cell_residual (Vector< double > &result, const FEValuesBase< dim > &fe, const ArrayView< const std::vector< Tensor< 1, dim >>> &input, double factor=1.) |
template<int dim> | |
void | nitsche_matrix (FullMatrix< double > &M, const FEValuesBase< dim > &fe, double penalty, double factor=1.) |
template<int dim> | |
void | nitsche_tangential_matrix (FullMatrix< double > &M, const FEValuesBase< dim > &fe, double penalty, double factor=1.) |
template<int dim> | |
void | nitsche_residual (Vector< double > &result, const FEValuesBase< dim > &fe, const std::vector< double > &input, const std::vector< Tensor< 1, dim >> &Dinput, const std::vector< double > &data, double penalty, double factor=1.) |
template<int dim> | |
void | nitsche_residual (Vector< double > &result, const FEValuesBase< dim > &fe, const ArrayView< const std::vector< double >> &input, const ArrayView< const std::vector< Tensor< 1, dim >>> &Dinput, const ArrayView< const std::vector< double >> &data, double penalty, double factor=1.) |
template<int dim> | |
void | ip_matrix (FullMatrix< double > &M11, FullMatrix< double > &M12, FullMatrix< double > &M21, FullMatrix< double > &M22, const FEValuesBase< dim > &fe1, const FEValuesBase< dim > &fe2, double penalty, double factor1=1., double factor2=-1.) |
template<int dim> | |
void | ip_tangential_matrix (FullMatrix< double > &M11, FullMatrix< double > &M12, FullMatrix< double > &M21, FullMatrix< double > &M22, const FEValuesBase< dim > &fe1, const FEValuesBase< dim > &fe2, double penalty, double factor1=1., double factor2=-1.) |
template<int dim> | |
void | ip_residual (Vector< double > &result1, Vector< double > &result2, const FEValuesBase< dim > &fe1, const FEValuesBase< dim > &fe2, const std::vector< double > &input1, const std::vector< Tensor< 1, dim >> &Dinput1, const std::vector< double > &input2, const std::vector< Tensor< 1, dim >> &Dinput2, double pen, double int_factor=1., double ext_factor=-1.) |
template<int dim> | |
void | ip_residual (Vector< double > &result1, Vector< double > &result2, const FEValuesBase< dim > &fe1, const FEValuesBase< dim > &fe2, const ArrayView< const std::vector< double >> &input1, const ArrayView< const std::vector< Tensor< 1, dim >>> &Dinput1, const ArrayView< const std::vector< double >> &input2, const ArrayView< const std::vector< Tensor< 1, dim >>> &Dinput2, double pen, double int_factor=1., double ext_factor=-1.) |
template<int dim, int spacedim, typename number > | |
double | compute_penalty (const MeshWorker::DoFInfo< dim, spacedim, number > &dinfo1, const MeshWorker::DoFInfo< dim, spacedim, number > &dinfo2, unsigned int deg1, unsigned int deg2) |
Local integrators related to the Laplacian and its DG formulations.
void LocalIntegrators::Laplace::cell_matrix | ( | FullMatrix< double > & | M, |
const FEValuesBase< dim > & | fe, | ||
const double | factor = 1. |
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Laplacian in weak form, namely on the cell Z the matrix
\[ \int_Z \nu \nabla u \cdot \nabla v \, dx. \]
The FiniteElement in fe
may be scalar or vector valued. In the latter case, the Laplacian is applied to each component separately.
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void LocalIntegrators::Laplace::nitsche_matrix | ( | FullMatrix< double > & | M, |
const FEValuesBase< dim > & | fe, | ||
double | penalty, | ||
double | factor = 1. |
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Weak boundary condition of Nitsche type for the Laplacian, namely on the face F the matrix
\[ \int_F \Bigl(\gamma u v - \partial_n u v - u \partial_n v\Bigr)\;ds. \]
Here, \(\gamma\) is the penalty
parameter suitably computed with compute_penalty().
void LocalIntegrators::Laplace::nitsche_tangential_matrix | ( | FullMatrix< double > & | M, |
const FEValuesBase< dim > & | fe, | ||
double | penalty, | ||
double | factor = 1. |
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Weak boundary condition of Nitsche type for the Laplacian applied to the tangential component only, namely on the face F the matrix
\[ \int_F \Bigl(\gamma u_\tau v_\tau - \partial_n u_\tau v_\tau - u_\tau \partial_n v_\tau\Bigr)\;ds. \]
Here, \(\gamma\) is the penalty
parameter suitably computed with compute_penalty().
void LocalIntegrators::Laplace::nitsche_residual | ( | Vector< double > & | result, |
const FEValuesBase< dim > & | fe, | ||
const std::vector< double > & | input, | ||
const std::vector< Tensor< 1, dim >> & | Dinput, | ||
const std::vector< double > & | data, | ||
double | penalty, | ||
double | factor = 1. |
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) |
Weak boundary condition for the Laplace operator by Nitsche, scalar version, namely on the face F the vector
\[ \int_F \Bigl(\gamma (u-g) v - \partial_n u v - (u-g) \partial_n v\Bigr)\;ds. \]
Here, u is the finite element function whose values and gradient are given in the arguments input
and Dinput
, respectively. g is the inhomogeneous boundary value in the argument data
. \(\gamma\) is the usual penalty parameter.
void LocalIntegrators::Laplace::nitsche_residual | ( | Vector< double > & | result, |
const FEValuesBase< dim > & | fe, | ||
const ArrayView< const std::vector< double >> & | input, | ||
const ArrayView< const std::vector< Tensor< 1, dim >>> & | Dinput, | ||
const ArrayView< const std::vector< double >> & | data, | ||
double | penalty, | ||
double | factor = 1. |
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) |
Weak boundary condition for the Laplace operator by Nitsche, vector valued version, namely on the face F the vector
\[ \int_F \Bigl(\gamma (\mathbf u- \mathbf g) \cdot \mathbf v - \partial_n \mathbf u \cdot \mathbf v - (\mathbf u-\mathbf g) \cdot \partial_n \mathbf v\Bigr)\;ds. \]
Here, u is the finite element function whose values and gradient are given in the arguments input
and Dinput
, respectively. g is the inhomogeneous boundary value in the argument data
. \(\gamma\) is the usual penalty parameter.
void LocalIntegrators::Laplace::ip_matrix | ( | FullMatrix< double > & | M11, |
FullMatrix< double > & | M12, | ||
FullMatrix< double > & | M21, | ||
FullMatrix< double > & | M22, | ||
const FEValuesBase< dim > & | fe1, | ||
const FEValuesBase< dim > & | fe2, | ||
double | penalty, | ||
double | factor1 = 1. , |
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double | factor2 = -1. |
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Flux for the interior penalty method for the Laplacian, namely on the face F the matrices associated with the bilinear form
\[ \int_F \Bigl( \gamma [u][v] - \{\nabla u\}[v\mathbf n] - [u\mathbf n]\{\nabla v\} \Bigr) \; ds. \]
The penalty parameter should always be the mean value of the penalties needed for stability on each side. In the case of constant coefficients, it can be computed using compute_penalty().
If factor2
is missing or negative, the factor is assumed the same on both sides. If factors differ, note that the penalty parameter has to be computed accordingly.
void LocalIntegrators::Laplace::ip_tangential_matrix | ( | FullMatrix< double > & | M11, |
FullMatrix< double > & | M12, | ||
FullMatrix< double > & | M21, | ||
FullMatrix< double > & | M22, | ||
const FEValuesBase< dim > & | fe1, | ||
const FEValuesBase< dim > & | fe2, | ||
double | penalty, | ||
double | factor1 = 1. , |
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double | factor2 = -1. |
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Flux for the interior penalty method for the Laplacian applied to the tangential components of a vector field, namely on the face F the matrices associated with the bilinear form
\[ \int_F \Bigl( \gamma [u_\tau][v_\tau] - \{\nabla u_\tau\}[v_\tau\mathbf n] - [u_\tau\mathbf n]\{\nabla v_\tau\} \Bigr) \; ds. \]
void LocalIntegrators::Laplace::ip_residual | ( | Vector< double > & | result1, |
Vector< double > & | result2, | ||
const FEValuesBase< dim > & | fe1, | ||
const FEValuesBase< dim > & | fe2, | ||
const std::vector< double > & | input1, | ||
const std::vector< Tensor< 1, dim >> & | Dinput1, | ||
const std::vector< double > & | input2, | ||
const std::vector< Tensor< 1, dim >> & | Dinput2, | ||
double | pen, | ||
double | int_factor = 1. , |
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double | ext_factor = -1. |
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void LocalIntegrators::Laplace::ip_residual | ( | Vector< double > & | result1, |
Vector< double > & | result2, | ||
const FEValuesBase< dim > & | fe1, | ||
const FEValuesBase< dim > & | fe2, | ||
const ArrayView< const std::vector< double >> & | input1, | ||
const ArrayView< const std::vector< Tensor< 1, dim >>> & | Dinput1, | ||
const ArrayView< const std::vector< double >> & | input2, | ||
const ArrayView< const std::vector< Tensor< 1, dim >>> & | Dinput2, | ||
double | pen, | ||
double | int_factor = 1. , |
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double | ext_factor = -1. |
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double LocalIntegrators::Laplace::compute_penalty | ( | const MeshWorker::DoFInfo< dim, spacedim, number > & | dinfo1, |
const MeshWorker::DoFInfo< dim, spacedim, number > & | dinfo2, | ||
unsigned int | deg1, | ||
unsigned int | deg2 | ||
) |
Auxiliary function computing the penalty parameter for interior penalty methods on rectangles.
Computation is done in two steps: first, we compute on each cell Zi the value Pi = pi(pi+1)/hi, where pi is the polynomial degree on cell Zi and hi is the length of Zi orthogonal to the current face.