This tutorial depends on step-12.
In this program, we use the interior penalty method and Nitsche's weak boundary conditions to solve Poisson's equation. We use multigrid methods on locally refined meshes, which are generated using a bulk criterion and a standard error estimator based on cell and face residuals. All operators are implemented using the MeshWorker interface.
Like in step-12, the discretization relies on finite element spaces, which are polynomial inside the mesh cells \(K\in \mathbb T_h\), but have no continuity between cells. Since such functions have two values on each interior face \(F\in \mathbb F_h^i\), one from each side, we define mean value and jump operators as follows: let K1 and K2 be the two cells sharing a face, and let the traces of functions ui and the outer normal vectors ni be labeled accordingly. Then, on the face, we let
\[
\average{ u } = \frac{u_1 + u_2}2
\]
Note, that if such an expression contains a normal vector, the averaging operator turns into a jump. The interior penalty method for the problem
\[
-\Delta u = f \text{ in }\Omega \qquad u = u^D \text{ on } \partial\Omega
\]
becomes
\begin{multline*}
\sum_{K\in \mathbb T_h} (\nabla u, \nabla v)_K
\\
+ \sum_{F \in F_h^i} \biggl\{4\sigma_F (\average{ u \mathbf n}, \average{ v \mathbf n })_F
- 2 (\average{ \nabla u },\average{ v\mathbf n })_F
- 2 (\average{ \nabla v },\average{ u\mathbf n })_F
\biggr\}
\\
+ \sum_{F \in F_h^b} \biggl\{2\sigma_F (u, v)_F
- (\partial_n u,v)_F
- (\partial_n v,u)_F
\biggr\}
\\
= (f, v)_\Omega + \sum_{F \in F_h^b} \biggl\{
2\sigma_F (u^D, v)_F - (\partial_n v,u^D)_F
\biggr\}.
\end{multline*}
Here, \(\sigma_F\) is the penalty parameter, which is chosen as follows: for a face F of a cell K, compute the value
\[
\sigma_{F,K} = p(p+1) \frac{|F|_{d-1}}{|K|_d},
\]
where p is the polynomial degree of the finite element functions and \(|\cdot|_d\) and \(|\cdot|_{d-1}\) denote the \(d\) and \(d-1\) dimensional Hausdorff measure of the corresponding object. If the face is at the boundary, choose \(\sigma_F = \sigma_{F,K}\). For an interior face, we take the average of the two values at this face.
In our finite element program, we distinguish three different integrals, corresponding to the sums over cells, interior faces and boundary faces above. Since the MeshWorker::loop organizes the sums for us, we only need to implement the integrals over each mesh element. The class MatrixIntegrator below has these three functions for the left hand side of the formula, the class RHSIntegrator for the right.
As we will see below, even the error estimate is of the same structure, since it can be written as
\begin{align*}
\eta^2 &= \eta_K^2 + \eta_F^2 + \eta_B^2
\\
\eta_K^2 &= \sum_{K\in \mathbb T_h} h^2 \|f + \Delta u_h\|^2
\\
\eta_F^2 &= \sum_{F \in F_h^i} \biggl\{
4 \sigma_F \| \average{u_h\mathbf n} \|^2 + h \|\average{\partial_n u_h}\|^2 \biggr\}
\\
\eta_B^2 &= \sum_{F \in F_h^b} 2\sigma_F \| u_h-u^D \|^2.
\end{align*}
Thus, the functions for assembling matrices, right hand side and error estimates below exhibit that these loops are all generic and can be programmed in the same way.
This program is related to step-12, in that it uses MeshWorker and discontinuous Galerkin methods. There we solved an advection problem, while here it is a diffusion problem. Here, we also use multigrid preconditioning and a theoretically justified error estimator, see Karakashian and Pascal (2003). The multilevel scheme was discussed in detail in Kanschat (2004). The adaptive iteration and its convergence have been discussed (for triangular meshes) in Hoppe, Kanschat, and Warburton (2009).
The commented program
The include files for the linear algebra: A regular SparseMatrix, which in turn will include the necessary files for SparsityPattern and Vector classes.
#include <deal.II/lac/sparse_matrix.h>
#include <deal.II/lac/dynamic_sparsity_pattern.h>
#include <deal.II/lac/solver_cg.h>
#include <deal.II/lac/precondition.h>
#include <deal.II/lac/precondition_block.h>
#include <deal.II/lac/block_vector.h>
Include files for setting up the mesh
#include <deal.II/grid/grid_generator.h>
#include <deal.II/grid/grid_refinement.h>
Include files for FiniteElement classes and DoFHandler.
#include <deal.II/fe/fe_q.h>
#include <deal.II/fe/fe_dgp.h>
#include <deal.II/fe/fe_dgq.h>
#include <deal.II/dofs/dof_tools.h>
#include <deal.II/fe/mapping_q1.h>
The include files for using the MeshWorker framework
#include <deal.II/meshworker/dof_info.h>
#include <deal.II/meshworker/integration_info.h>
#include <deal.II/meshworker/assembler.h>
#include <deal.II/meshworker/loop.h>
The include file for local integrators associated with the Laplacian
#include <deal.II/integrators/laplace.h>
Support for multigrid methods
#include <deal.II/multigrid/mg_tools.h>
#include <deal.II/multigrid/multigrid.h>
#include <deal.II/multigrid/mg_matrix.h>
#include <deal.II/multigrid/mg_transfer.h>
#include <deal.II/multigrid/mg_coarse.h>
#include <deal.II/multigrid/mg_smoother.h>
Finally, we take our exact solution from the library as well as quadrature and additional tools.
#include <deal.II/base/function_lib.h>
#include <deal.II/base/quadrature_lib.h>
#include <deal.II/numerics/vector_tools.h>
#include <deal.II/numerics/data_out.h>
#include <iostream>
#include <fstream>
All classes of the deal.II library are in the namespace dealii. In order to save typing, we tell the compiler to search names in there as well.
This is the function we use to set the boundary values and also the exact solution we compare to.
The local integrators
The MeshWorker::loop() function separates what needs to be done for local integration, from the loops over cells and faces. It does this by calling functions that integrate over a cell, a boundary face, or an interior face, and letting them create the local contributions and then in a separate step calling a function that moves these local contributions into the global objects. We will use this approach for computing the matrices, the right hand side, the error estimator, and the actual error computation in the functions below. For each of these operations, we provide a namespace that contains a set of functions for cell, boundary, and interior face contributions.
All the information needed for these local integration is provided by MeshWorker::DoFInfo<dim> and MeshWorker::IntegrationInfo<dim>. In each case, the functions' signatures is fixed: MeshWorker::loop() wants to call functions with a specific set of arguments, so the signature of the functions cannot be changed.
The first namespace defining local integrators is responsible for assembling the global matrix as well as the level matrices. On each cell, we integrate the Dirichlet form as well as the Nitsche boundary conditions and the interior penalty fluxes between cells.
The boundary and flux terms need a penalty parameter, which should be adjusted to the cell size and the polynomial degree. We compute it in two steps: First, we compute on each cell \(K_i\) the value \(P_i = p_i(p_i+1)/h_i\), where \(p_i\) is the polynomial degree on cell \(K_i\) and \(h_i\) is the length of \(K_i\) orthogonal to the current face. Second, if exactly one of the two cells adjacent to the face has children, its penalty is multiplied by two (to account for the fact that the mesh size \(h_i\) there is only half that previously computed); it is possible that both adjacent cells are refined, in which case we are integrating over a non-active face and no adjustment is necessary. Finally, we return the average of the two penalty values.
namespace MatrixIntegrator
{
template <int dim>
unsigned int deg1,
unsigned int deg2)
{
const unsigned int normal1 =
const unsigned int normal2 =
const unsigned int deg1sq = (deg1 == 0) ? 1 : deg1 * (deg1 + 1);
const unsigned int deg2sq = (deg2 == 0) ? 1 : deg2 * (deg2 + 1);
double penalty1 = deg1sq / dinfo1.
cell->extent_in_direction(normal1);
double penalty2 = deg2sq / dinfo2.cell->extent_in_direction(normal2);
if (dinfo1.
cell->has_children() && !dinfo2.cell->has_children())
penalty1 *= 2;
else if (!dinfo1.
cell->has_children() && dinfo2.cell->has_children())
penalty2 *= 2;
const double penalty = 0.5 * (penalty1 + penalty2);
return penalty;
}
template <int dim>
{
for (unsigned int k = 0; k < info.fe_values().n_quadrature_points; ++k)
{
const double dx = info.fe_values().JxW(k);
for (unsigned int i = 0; i < info.fe_values().dofs_per_cell; ++i)
{
const double Mii = (info.fe_values().shape_grad(i, k) *
info.fe_values().shape_grad(i, k) * dx);
M(i, i) += Mii;
for (unsigned int j = i + 1; j < info.fe_values().dofs_per_cell;
++j)
{
const double Mij = info.fe_values().shape_grad(j, k) *
info.fe_values().shape_grad(i, k) * dx;
M(i, j) += Mij;
M(j, i) += Mij;
}
}
}
}
Triangulation< dim, spacedim >::cell_iterator cell
The current cell.
MatrixBlock< FullMatrix< number > > & matrix(const unsigned int i, const bool external=false)
Boundary faces use the Nitsche method to impose boundary values:
template <int dim>
{
const unsigned int polynomial_degree =
info.fe_values(0).get_fe().tensor_degree();
const double ip_penalty =
ip_penalty_factor(dinfo, dinfo, polynomial_degree, polynomial_degree);
for (unsigned int k = 0; k < fe_face_values.n_quadrature_points; ++k)
{
const double dx = fe_face_values.JxW(k);
for (unsigned int i = 0; i < fe_face_values.dofs_per_cell; ++i)
for (unsigned int j = 0; j < fe_face_values.dofs_per_cell; ++j)
M(i, j) += (2. * fe_face_values.shape_value(i, k) * ip_penalty *
fe_face_values.shape_value(j, k) -
(n * fe_face_values.shape_grad(i, k)) *
fe_face_values.shape_value(j, k) -
(n * fe_face_values.shape_grad(j, k)) *
fe_face_values.shape_value(i, k)) *
dx;
}
}
#define AssertDimension(dim1, dim2)
Interior faces use the interior penalty method:
template <int dim>
{
const unsigned int polynomial_degree =
info1.fe_values(0).get_fe().tensor_degree();
const double ip_penalty =
ip_penalty_factor(dinfo1, dinfo2, polynomial_degree, polynomial_degree);
const double nui = 1.;
const double nue = 1.;
const double nu = .5 * (nui + nue);
for (unsigned int k = 0; k < fe_face_values_1.n_quadrature_points; ++k)
{
const double dx = fe_face_values_1.JxW(k);
for (unsigned int i = 0; i < fe_face_values_1.dofs_per_cell; ++i)
{
for (unsigned int j = 0; j < fe_face_values_1.dofs_per_cell; ++j)
{
const double vi = fe_face_values_1.shape_value(i, k);
const double dnvi = n * fe_face_values_1.shape_grad(i, k);
const double ve = fe_face_values_2.shape_value(i, k);
const double dnve = n * fe_face_values_2.shape_grad(i, k);
const double ui = fe_face_values_1.shape_value(j, k);
const double dnui = n * fe_face_values_1.shape_grad(j, k);
const double ue = fe_face_values_2.shape_value(j, k);
const double dnue = n * fe_face_values_2.shape_grad(j, k);
M11(i, j) += (-.5 * nui * dnvi * ui - .5 * nui * dnui * vi +
nu * ip_penalty * ui * vi) *
dx;
M12(i, j) += (.5 * nui * dnvi * ue - .5 * nue * dnue * vi -
nu * ip_penalty * vi * ue) *
dx;
M21(i, j) += (-.5 * nue * dnve * ui + .5 * nui * dnui * ve -
nu * ip_penalty * ui * ve) *
dx;
M22(i, j) += (.5 * nue * dnve * ue + .5 * nue * dnue * ve +
nu * ip_penalty * ue * ve) *
dx;
}
}
}
}
}
The second set of local integrators builds the right hand side. In our example, the right hand side function is zero, such that only the boundary condition is set here in weak form.
namespace RHSIntegrator
{
template <int dim>
{}
template <int dim>
{
std::vector<double> boundary_values(fe.n_quadrature_points);
exact_solution.value_list(fe.get_quadrature_points(), boundary_values);
const unsigned int degree = fe.get_fe().tensor_degree();
const double penalty = 2. * degree * (degree + 1) *
dinfo.
face->measure() / dinfo.
cell->measure();
for (unsigned k = 0; k < fe.n_quadrature_points; ++k)
for (unsigned int i = 0; i < fe.dofs_per_cell; ++i)
local_vector(i) +=
(-penalty * fe.shape_value(i, k)
+
fe.normal_vector(k) * fe.shape_grad(i, k))
* boundary_values[k] * fe.JxW(k);
}
template <int dim>
{}
}
BlockType & block(const unsigned int i)
Triangulation< dim, spacedim >::face_iterator face
The current face.
BlockVector< number > & vector(const unsigned int i)
The third local integrator is responsible for the contributions to the error estimate. This is the standard energy estimator due to Karakashian and Pascal (2003). The cell contribution is the Laplacian of the discrete solution, since the right hand side is zero.
namespace Estimator
{
template <int dim>
{
const std::vector<Tensor<2, dim>> &DDuh = info.hessians[0][0];
for (unsigned k = 0; k < fe.n_quadrature_points; ++k)
{
const double t = dinfo.
cell->diameter() *
trace(DDuh[k]);
dinfo.
value(0) += t * t * fe.JxW(k);
}
}
number & value(const unsigned int i)
::VectorizedArray< Number, width > sqrt(const ::VectorizedArray< Number, width > &)
DEAL_II_HOST constexpr Number trace(const SymmetricTensor< 2, dim2, Number > &)
At the boundary, we use simply a weighted form of the boundary residual, namely the norm of the difference between the finite element solution and the correct boundary condition.
template <int dim>
{
std::vector<double> boundary_values(fe.n_quadrature_points);
exact_solution.value_list(fe.get_quadrature_points(), boundary_values);
const std::vector<double> &uh = info.values[0][0];
const unsigned int degree = fe.get_fe().tensor_degree();
const double penalty = 2. * degree * (degree + 1) *
dinfo.
face->measure() / dinfo.
cell->measure();
for (unsigned k = 0; k < fe.n_quadrature_points; ++k)
{
const double diff = boundary_values[k] - uh[k];
dinfo.
value(0) += penalty * diff * diff * fe.JxW(k);
}
}
Finally, on interior faces, the estimator consists of the jumps of the solution and its normal derivative, weighted appropriately.
template <int dim>
{
const std::vector<double> &uh1 = info1.values[0][0];
const std::vector<double> &uh2 = info2.values[0][0];
const std::vector<Tensor<1, dim>> &Duh1 = info1.gradients[0][0];
const std::vector<Tensor<1, dim>> &Duh2 = info2.gradients[0][0];
const double penalty1 =
degree * (degree + 1) * dinfo1.
face->measure() / dinfo1.
cell->measure();
const double penalty2 =
degree * (degree + 1) * dinfo2.face->measure() / dinfo2.cell->measure();
const double penalty = penalty1 + penalty2;
const double h = dinfo1.
face->measure();
for (unsigned k = 0; k < fe.n_quadrature_points; ++k)
{
const double diff1 = uh1[k] - uh2[k];
const double diff2 =
fe.normal_vector(k) * Duh1[k] - fe.normal_vector(k) * Duh2[k];
(penalty * diff1 * diff1 + h * diff2 * diff2) * fe.JxW(k);
}
dinfo2.value(0) = dinfo1.
value(0);
}
}
const FiniteElement< dim, spacedim > & get_fe() const
unsigned int tensor_degree() const
Finally we have an integrator for the error. Since the energy norm for discontinuous Galerkin problems not only involves the difference of the gradient inside the cells, but also the jump terms across faces and at the boundary, we cannot just use VectorTools::integrate_difference(). Instead, we use the MeshWorker interface to compute the error ourselves.
There are several different ways to define this energy norm, but all of them are equivalent to each other uniformly with mesh size (some not uniformly with polynomial degree). Here, we choose
\[ \|u\|_{1,h} =
\sum_{K\in \mathbb T_h} \|\nabla u\|_K^2 + \sum_{F \in F_h^i}
4\sigma_F\|\average{ u \mathbf n}\|^2_F + \sum_{F \in F_h^b}
2\sigma_F\|u\|^2_F \]
Below, the first function is, as always, the integration on cells. There is currently no good interface in MeshWorker that would allow us to access values of regular functions in the quadrature points. Thus, we have to create the vectors for the exact function's values and gradients inside the cell integrator. After that, everything is as before and we just add up the squares of the differences.
Additionally to computing the error in the energy norm, we use the capability of the mesh worker to compute two functionals at the same time and compute the L2-error in the same loop. Obviously, this one does not have any jump terms and only appears in the integration on cells.
namespace ErrorIntegrator
{
template <int dim>
{
std::vector<Tensor<1, dim>> exact_gradients(fe.n_quadrature_points);
std::vector<double> exact_values(fe.n_quadrature_points);
exact_solution.gradient_list(fe.get_quadrature_points(), exact_gradients);
exact_solution.value_list(fe.get_quadrature_points(), exact_values);
const std::vector<Tensor<1, dim>> &Duh = info.gradients[0][0];
const std::vector<double> &uh = info.values[0][0];
for (unsigned k = 0; k < fe.n_quadrature_points; ++k)
{
double sum = 0;
for (unsigned int d = 0; d < dim; ++d)
{
const double diff = exact_gradients[k][d] - Duh[k][d];
sum += diff * diff;
}
const double diff = exact_values[k] - uh[k];
dinfo.
value(0) += sum * fe.JxW(k);
dinfo.
value(1) += diff * diff * fe.JxW(k);
}
}
template <int dim>
{
std::vector<double> exact_values(fe.n_quadrature_points);
exact_solution.value_list(fe.get_quadrature_points(), exact_values);
const std::vector<double> &uh = info.values[0][0];
const unsigned int degree = fe.get_fe().tensor_degree();
const double penalty = 2. * degree * (degree + 1) *
dinfo.
face->measure() / dinfo.
cell->measure();
for (unsigned k = 0; k < fe.n_quadrature_points; ++k)
{
const double diff = exact_values[k] - uh[k];
dinfo.
value(0) += penalty * diff * diff * fe.JxW(k);
}
}
template <int dim>
{
const std::vector<double> &uh1 = info1.values[0][0];
const std::vector<double> &uh2 = info2.values[0][0];
const double penalty1 =
degree * (degree + 1) * dinfo1.
face->measure() / dinfo1.
cell->measure();
const double penalty2 =
degree * (degree + 1) * dinfo2.face->measure() / dinfo2.cell->measure();
const double penalty = penalty1 + penalty2;
for (unsigned k = 0; k < fe.n_quadrature_points; ++k)
{
const double diff = uh1[k] - uh2[k];
dinfo1.
value(0) += (penalty * diff * diff) * fe.JxW(k);
}
dinfo2.value(0) = dinfo1.
value(0);
}
}
The main class
This class does the main job, like in previous examples. For a description of the functions declared here, please refer to the implementation below.
template <int dim>
class InteriorPenaltyProblem
{
public:
InteriorPenaltyProblem();
void run(unsigned int n_steps);
private:
void setup_system();
void assemble_matrix();
void assemble_mg_matrix();
void assemble_right_hand_side();
void error();
double estimate();
void solve();
void output_results(const unsigned int cycle) const;
The member objects related to the discretization are here.
const ::parallel::distributed::Triangulation< dim, spacedim > * triangulation
Then, we have the matrices and vectors related to the global discrete system.
Finally, we have a group of sparsity patterns and sparse matrices related to the multilevel preconditioner. First, we have a level matrix and its sparsity pattern.
When we perform multigrid with local smoothing on locally refined meshes, additional matrices are required; see Kanschat (2004). Here is the sparsity pattern for these edge matrices. We only need one, because the pattern of the up matrix is the transpose of that of the down matrix. Actually, we do not care too much about these details, since the MeshWorker is filling these matrices.
The flux matrix at the refinement edge, coupling fine level degrees of freedom to coarse level.
The transpose of the flux matrix at the refinement edge, coupling coarse level degrees of freedom to fine level.
The constructor simply sets up the coarse grid and the DoFHandler.
template <int dim>
InteriorPenaltyProblem<dim>::InteriorPenaltyProblem()
, mapping()
, fe(3)
, estimates(1)
{
}
void hyper_cube_slit(Triangulation< dim > &tria, const double left=0., const double right=1., const bool colorize=false)
In this function, we set up the dimension of the linear system and the sparsity patterns for the global matrix as well as the level matrices.
template <int dim>
void InteriorPenaltyProblem<dim>::setup_system()
{
First, we use the finite element to distribute degrees of freedom over the mesh and number them.
dof_handler.distribute_dofs(fe);
dof_handler.distribute_mg_dofs();
unsigned int n_dofs = dof_handler.n_dofs();
Then, we already know the size of the vectors representing finite element functions.
solution.reinit(n_dofs);
right_hand_side.reinit(n_dofs);
Next, we set up the sparsity pattern for the global matrix. Since we do not know the row sizes in advance, we first fill a temporary DynamicSparsityPattern object and copy it to the regular SparsityPattern once it is complete.
sparsity.copy_from(dsp);
matrix.reinit(sparsity);
unsigned int n_levels() const
void make_flux_sparsity_pattern(const DoFHandler< dim, spacedim > &dof_handler, SparsityPatternBase &sparsity_pattern)
The global system is set up, now we attend to the level matrices. We resize all matrix objects to hold one matrix per level.
mg_matrix.resize(0, n_levels - 1);
mg_matrix.clear_elements();
mg_matrix_dg_up.resize(0, n_levels - 1);
mg_matrix_dg_up.clear_elements();
mg_matrix_dg_down.resize(0, n_levels - 1);
mg_matrix_dg_down.clear_elements();
It is important to update the sparsity patterns after clear() was called for the level matrices, since the matrices lock the sparsity pattern through the SmartPointer and Subscriptor mechanism.
mg_sparsity.resize(0, n_levels - 1);
mg_sparsity_dg_interface.resize(0, n_levels - 1);
Now all objects are prepared to hold one sparsity pattern or matrix per level. What's left is setting up the sparsity patterns on each level.
for (
unsigned int level = mg_sparsity.min_level();
level <= mg_sparsity.max_level();
{
These are roughly the same lines as above for the global matrix, now for each level.
mg_sparsity[
level].copy_from(dsp);
Additionally, we need to initialize the transfer matrices at the refinement edge between levels. They are stored at the index referring to the finer of the two indices, thus there is no such object on level 0.
{
dof_handler.n_dofs(
level));
mg_sparsity_dg_interface[
level].copy_from(dsp);
mg_matrix_dg_up[
level].reinit(mg_sparsity_dg_interface[
level]);
mg_matrix_dg_down[
level].reinit(mg_sparsity_dg_interface[
level]);
}
}
}
void reinit(const size_type m, const size_type n, const IndexSet &rowset=IndexSet())
In this function, we assemble the global system matrix, where by global we indicate that this is the matrix of the discrete system we solve and it is covering the whole mesh.
template <int dim>
void InteriorPenaltyProblem<dim>::assemble_matrix()
{
First, we need t set up the object providing the values we integrate. This object contains all FEValues and FEFaceValues objects needed and also maintains them automatically such that they always point to the current cell. To this end, we need to tell it first, where and what to compute. Since we are not doing anything fancy, we can rely on their standard choice for quadrature rules.
Since their default update flags are minimal, we add what we need additionally, namely the values and gradients of shape functions on all objects (cells, boundary and interior faces). Afterwards, we are ready to initialize the container, which will create all necessary FEValuesBase objects for integration.
info_box.initialize(fe, mapping);
void add_update_flags_all(const UpdateFlags flags)
@ update_values
Shape function values.
@ update_gradients
Shape function gradients.
This is the object into which we integrate local data. It is filled by the local integration routines in MatrixIntegrator and then used by the assembler to distribute the information into the global matrix.
Furthermore, we need an object that assembles the local matrix into the global matrix. These assembler objects have all the knowledge of the structures of the target object, in this case a SparseMatrix, possible constraints and the mesh structure.
void initialize(MatrixType &m)
Now, we throw everything into a MeshWorker::loop<dim, dim>(), which here traverses all active cells of the mesh, computes cell and face matrices and assembles them into the global matrix. We use the variable dof_handler here in order to use the global numbering of degrees of freedom.
dof_handler.end(),
dof_info,
info_box,
&MatrixIntegrator::cell<dim>,
&MatrixIntegrator::boundary<dim>,
&MatrixIntegrator::face<dim>,
assembler);
}
void loop(IteratorType begin, std_cxx20::type_identity_t< IteratorType > end, DOFINFO &dinfo, INFOBOX &info, const std::function< void(std_cxx20::type_identity_t< DOFINFO > &, typename INFOBOX::CellInfo &)> &cell_worker, const std::function< void(std_cxx20::type_identity_t< DOFINFO > &, typename INFOBOX::CellInfo &)> &boundary_worker, const std::function< void(std_cxx20::type_identity_t< DOFINFO > &, std_cxx20::type_identity_t< DOFINFO > &, typename INFOBOX::CellInfo &, typename INFOBOX::CellInfo &)> &face_worker, AssemblerType &assembler, const LoopControl &lctrl=LoopControl())
Now, we do the same for the level matrices. Not too surprisingly, this function looks like a twin of the previous one. Indeed, there are only two minor differences.
template <int dim>
void InteriorPenaltyProblem<dim>::assemble_mg_matrix()
{
info_box.initialize(fe, mapping);
Obviously, the assembler needs to be replaced by one filling level matrices. Note that it automatically fills the edge matrices as well.
assembler.initialize_fluxes(mg_matrix_dg_up, mg_matrix_dg_down);
void initialize(MGLevelObject< MatrixType > &m)
Here is the other difference to the previous function: we run over all cells, not only the active ones. And we use functions ending on _mg since we need the degrees of freedom on each level, not the global numbering.
dof_handler.end_mg(),
dof_info,
info_box,
&MatrixIntegrator::cell<dim>,
&MatrixIntegrator::boundary<dim>,
&MatrixIntegrator::face<dim>,
assembler);
}
Here we have another clone of the assemble function. The difference to assembling the system matrix consists in that we assemble a vector here.
template <int dim>
void InteriorPenaltyProblem<dim>::assemble_right_hand_side()
{
info_box.initialize(fe, mapping);
@ update_quadrature_points
Transformed quadrature points.
Since this assembler allows us to fill several vectors, the interface is a little more complicated as above. The pointers to the vectors have to be stored in an AnyData object. While this seems to cause two extra lines of code here, it actually comes handy in more complex applications.
assembler.initialize(data);
dof_handler.end(),
dof_info,
info_box,
&RHSIntegrator::cell<dim>,
&RHSIntegrator::boundary<dim>,
&RHSIntegrator::face<dim>,
assembler);
right_hand_side *= -1.;
}
void add(type entry, const std::string &name)
Add a new data object.
Now that we have coded all functions building the discrete linear system, it is about time that we actually solve it.
template <int dim>
void InteriorPenaltyProblem<dim>::solve()
{
The solver of choice is conjugate gradient.
Now we are setting up the components of the multilevel preconditioner. First, we need transfer between grid levels. The object we are using here generates sparse matrices for these transfers.
mg_transfer.
build(dof_handler);
void build(const DoFHandler< dim, spacedim > &dof_handler)
Then, we need an exact solver for the matrix on the coarsest level.
void copy_from(const MatrixType &)
void initialize(const FullMatrix< number > &A)
While transfer and coarse grid solver are pretty much generic, more flexibility is offered for the smoother. First, we choose Gauss-Seidel as our smoothing method.
RELAXATION::AdditionalData smoother_data(1.);
mg_smoother.initialize(mg_matrix, smoother_data);
Do two smoothing steps on each level.
mg_smoother.set_steps(2);
Since the SOR method is not symmetric, but we use conjugate gradient iteration below, here is a trick to make the multilevel preconditioner a symmetric operator even for nonsymmetric smoothers.
mg_smoother.set_symmetric(true);
The smoother class optionally implements the variable V-cycle, which we do not want here.
mg_smoother.set_variable(false);
Finally, we must wrap our matrices in an object having the required multiplication functions.
Now, we are ready to set up the V-cycle operator and the multilevel preconditioner.
mgmatrix, mg_coarse, mg_transfer, mg_smoother, mg_smoother);
Let us not forget the edge matrices needed because of the adaptive refinement.
mg.set_edge_flux_matrices(mgdown, mgup);
After all preparations, wrap the Multigrid object into another object, which can be used as a regular preconditioner,
preconditioner(dof_handler,
mg, mg_transfer);
and use it to solve the system.
solver.solve(matrix, solution, right_hand_side, preconditioner);
}
Another clone of the assemble function. The big difference to the previous ones is here that we also have an input vector.
template <int dim>
double InteriorPenaltyProblem<dim>::estimate()
{
The results of the estimator are stored in a vector with one entry per cell. Since cells in deal.II are not numbered, we have to create our own numbering in order to use this vector. For the assembler used below the information in which component of a vector the result is stored is transmitted by the user_index variable for each cell. We need to set this numbering up here.
On the other hand, somebody might have used the user indices already. So, let's be good citizens and save them before tampering with them.
std::vector<unsigned int> old_user_indices;
unsigned int i = 0;
cell->set_user_index(i++);
unsigned int n_active_cells() const
void save_user_indices(std::vector< unsigned int > &v) const
This starts like before,
const unsigned int n_gauss_points =
dof_handler.get_fe().tensor_degree() + 1;
n_gauss_points + 1,
n_gauss_points);
void initialize_gauss_quadrature(unsigned int n_cell_points, unsigned int n_boundary_points, unsigned int n_face_points, const bool force=true)
but now we need to notify the info box of the finite element function we want to evaluate in the quadrature points. First, we create an AnyData object with this vector, which is the solution we just computed.
Then, we tell the Meshworker::VectorSelector for cells, that we need the second derivatives of this solution (to compute the Laplacian). Therefore, the Boolean arguments selecting function values and first derivatives a false, only the last one selecting second derivatives is true.
info_box.cell_selector.add("solution", false, false, true);
On interior and boundary faces, we need the function values and the first derivatives, but not second derivatives.
info_box.boundary_selector.add("solution", true, true, false);
info_box.face_selector.add("solution", true, true, false);
And we continue as before, with the exception that the default update flags are already adjusted to the values and derivatives we requested above.
info_box.initialize(fe, mapping, solution_data, solution);
The assembler stores one number per cell, but else this is the same as in the computation of the right hand side.
assembler.initialize(out_data, false);
dof_handler.end(),
dof_info,
info_box,
&Estimator::cell<dim>,
&Estimator::boundary<dim>,
&Estimator::face<dim>,
assembler);
Right before we return the result of the error estimate, we restore the old user indices.
return estimates.
block(0).l2_norm();
}
void load_user_indices(const std::vector< unsigned int > &v)
Here we compare our finite element solution with the (known) exact solution and compute the mean quadratic error of the gradient and the function itself. This function is a clone of the estimation function right above.
Since we compute the error in the energy and the L2-norm, respectively, our block vector needs two blocks here.
template <int dim>
void InteriorPenaltyProblem<dim>::error()
{
std::vector<unsigned int> old_user_indices;
unsigned int i = 0;
cell->set_user_index(i++);
const unsigned int n_gauss_points =
dof_handler.get_fe().tensor_degree() + 1;
n_gauss_points + 1,
n_gauss_points);
info_box.cell_selector.
add(
"solution",
true,
true,
false);
info_box.boundary_selector.add("solution", true, false, false);
info_box.face_selector.add("solution", true, false, false);
info_box.initialize(fe, mapping, solution_data, solution);
assembler.initialize(out_data, false);
dof_handler.end(),
dof_info,
info_box,
&ErrorIntegrator::cell<dim>,
&ErrorIntegrator::boundary<dim>,
&ErrorIntegrator::face<dim>,
assembler);
deallog <<
"energy-error: " << errors.
block(0).l2_norm() << std::endl;
deallog <<
"L2-error: " << errors.
block(1).l2_norm() << std::endl;
}
void add(const std::vector< size_type > &indices, const std::vector< OtherNumber > &values)
Create graphical output. We produce the filename by collating the name from its various components, including the refinement cycle that we output with two digits.
template <int dim>
void
InteriorPenaltyProblem<dim>::output_results(const unsigned int cycle) const
{
const std::string filename =
deallog <<
"Writing solution to <" << filename <<
">..." << std::endl
<< std::endl;
std::ofstream gnuplot_output(filename);
data_out.add_data_vector(solution, "u");
data_out.add_data_vector(estimates.
block(0),
"est");
data_out.build_patches();
data_out.write_gnuplot(gnuplot_output);
}
void attach_dof_handler(const DoFHandler< dim, spacedim > &)
std::string int_to_string(const unsigned int value, const unsigned int digits=numbers::invalid_unsigned_int)
And finally the adaptive loop, more or less like in previous examples.
template <int dim>
void InteriorPenaltyProblem<dim>::run(unsigned int n_steps)
{
deallog <<
"Element: " << fe.get_name() << std::endl;
for (unsigned int s = 0; s < n_steps; ++s)
{
deallog <<
"Step " << s << std::endl;
if (estimates.
block(0).size() == 0)
else
{
}
<< std::endl;
setup_system();
deallog <<
"DoFHandler " << dof_handler.n_dofs() <<
" dofs, level dofs";
deallog <<
' ' << dof_handler.n_dofs(l);
deallog <<
"Assemble matrix" << std::endl;
assemble_matrix();
deallog <<
"Assemble multilevel matrix" << std::endl;
assemble_mg_matrix();
deallog <<
"Assemble right hand side" << std::endl;
assemble_right_hand_side();
solve();
error();
deallog <<
"Estimate " << estimate() << std::endl;
output_results(s);
}
}
}
int main()
{
try
{
using namespace Step39;
std::ofstream logfile("deallog");
InteriorPenaltyProblem<2> test1;
test1.run(12);
}
catch (std::exception &exc)
{
std::cerr << std::endl
<< std::endl
<< "----------------------------------------------------"
<< std::endl;
std::cerr << "Exception on processing: " << std::endl
<< exc.what() << std::endl
<< "Aborting!" << std::endl
<< "----------------------------------------------------"
<< std::endl;
return 1;
}
catch (...)
{
std::cerr << std::endl
<< std::endl
<< "----------------------------------------------------"
<< std::endl;
std::cerr << "Unknown exception!" << std::endl
<< "Aborting!" << std::endl
<< "----------------------------------------------------"
<< std::endl;
return 1;
}
return 0;
}
void attach(std::ostream &o, const bool print_job_id=true, const std::ios_base::fmtflags flags=std::ios::showpoint|std::ios::left)
unsigned int depth_console(const unsigned int n)
void refine_global(const unsigned int times=1)
virtual void execute_coarsening_and_refinement() override
void refine_and_coarsen_fixed_fraction(Triangulation< dim, spacedim > &tria, const Vector< Number > &criteria, const double top_fraction, const double bottom_fraction, const unsigned int max_n_cells=std::numeric_limits< unsigned int >::max(), const VectorTools::NormType norm_type=VectorTools::L1_norm)
Results
Logfile output
First, the program produces the usual logfile here stored in deallog. It reads (with omission of intermediate steps)
DEAL::Step 0
DEAL::Assemble matrix
DEAL::Assemble multilevel matrix
DEAL::Assemble right hand side
DEAL::Solve
DEAL:cg::Starting value 37.4071
DEAL:cg::Convergence step 13 value 1.64974e-13
DEAL::energy-error: 0.297419
DEAL::L2-error: 0.00452447
DEAL::Estimate 0.990460
DEAL::Writing solution to <sol-00.gnuplot>...
DEAL::
DEAL::Step 1
DEAL::Assemble matrix
DEAL::Assemble multilevel matrix
DEAL::Assemble right hand side
DEAL::Solve
DEAL:cg::Starting value 37.4071
DEAL:cg::Convergence step 14 value 3.72262e-13
DEAL::energy-error: 0.258559
DEAL::L2-error: 0.00288510
DEAL::Estimate 0.738624
DEAL::Writing solution to <sol-01.gnuplot>...
DEAL::
DEAL::Step 2
DEAL::Assemble matrix
DEAL::Assemble multilevel matrix
DEAL::Assemble right hand side
DEAL::Solve
DEAL:cg::Starting value 37.4071
DEAL:cg::Convergence step 15 value 1.91610e-13
DEAL::energy-error: 0.189234
DEAL::L2-error: 0.00147954
DEAL::Estimate 0.657507
DEAL::Writing solution to <sol-02.gnuplot>...
...
DEAL::Step 10
DEAL::
DoFHandler 3712 dofs,
level dofs 64 256 896 768 768 640 512 256 256 256 256
DEAL::Assemble matrix
DEAL::Assemble multilevel matrix
DEAL::Assemble right hand side
DEAL::Solve
DEAL:cg::Starting value 51.1571
DEAL:cg::Convergence step 15 value 7.19599e-13
DEAL::energy-error: 0.0132475
DEAL::L2-error: 1.00423e-05
DEAL::Estimate 0.0470724
DEAL::Writing solution to <sol-10.gnuplot>...
DEAL::
DEAL::Step 11
DEAL::
DoFHandler 5152 dofs,
level dofs 64 256 1024 1024 896 768 768 640 448 320 320 320
DEAL::Assemble matrix
DEAL::Assemble multilevel matrix
DEAL::Assemble right hand side
DEAL::Solve
DEAL:cg::Starting value 52.2226
DEAL:cg::Convergence step 15 value 8.15195e-13
DEAL::energy-error: 0.00934891
DEAL::L2-error: 5.41095e-06
DEAL::Estimate 0.0329102
DEAL::Writing solution to <sol-11.gnuplot>...
DEAL::
This log for instance shows that the number of conjugate gradient iteration steps is constant at approximately 15.
Postprocessing of the logfile
Using the perl script postprocess.pl, we extract relevant data into output.dat, which can be used to plot graphs with gnuplot. The graph above for instance was produced using the gnuplot script plot_errors.gpl via
./step-39 | perl postprocess.pl >output.dat
gnuplot plot_errors.gpl
Reference data can be found in output.reference.dat.
The plain program
#include <iostream>
#include <fstream>
namespace Step39
{
namespace MatrixIntegrator
{
template <int dim>
unsigned int deg1,
unsigned int deg2)
{
const unsigned int normal1 =
const unsigned int normal2 =
const unsigned int deg1sq = (deg1 == 0) ? 1 : deg1 * (deg1 + 1);
const unsigned int deg2sq = (deg2 == 0) ? 1 : deg2 * (deg2 + 1);
double penalty1 = deg1sq / dinfo1.
cell->extent_in_direction(normal1);
double penalty2 = deg2sq / dinfo2.
cell->extent_in_direction(normal2);
if (dinfo1.
cell->has_children() && !dinfo2.
cell->has_children())
penalty1 *= 2;
else if (!dinfo1.
cell->has_children() && dinfo2.
cell->has_children())
penalty2 *= 2;
const double penalty = 0.5 * (penalty1 + penalty2);
return penalty;
}
template <int dim>
{
for (
unsigned int k = 0; k < info.
fe_values().n_quadrature_points; ++k)
{
for (
unsigned int i = 0; i < info.
fe_values().dofs_per_cell; ++i)
{
const double Mii = (info.
fe_values().shape_grad(i, k) *
M(i, i) += Mii;
for (
unsigned int j = i + 1; j < info.
fe_values().dofs_per_cell;
++j)
{
const double Mij = info.
fe_values().shape_grad(j, k) *
M(i, j) += Mij;
M(j, i) += Mij;
}
}
}
}
template <int dim>
{
const unsigned int polynomial_degree =
const double ip_penalty =
ip_penalty_factor(dinfo, dinfo, polynomial_degree, polynomial_degree);
{
const double dx = fe_face_values.
JxW(k);
M(i, j) += (2. * fe_face_values.
shape_value(i, k) * ip_penalty *
dx;
}
}
template <int dim>
{
const unsigned int polynomial_degree =
const double ip_penalty =
ip_penalty_factor(dinfo1, dinfo2, polynomial_degree, polynomial_degree);
const double nui = 1.;
const double nue = 1.;
const double nu = .5 * (nui + nue);
{
const double dx = fe_face_values_1.
JxW(k);
for (
unsigned int i = 0; i < fe_face_values_1.
dofs_per_cell; ++i)
{
for (
unsigned int j = 0; j < fe_face_values_1.
dofs_per_cell; ++j)
{
const double dnvi = n * fe_face_values_1.
shape_grad(i, k);
const double dnve = n * fe_face_values_2.
shape_grad(i, k);
const double dnui = n * fe_face_values_1.
shape_grad(j, k);
const double dnue = n * fe_face_values_2.
shape_grad(j, k);
M11(i, j) += (-.5 * nui * dnvi * ui - .5 * nui * dnui * vi +
nu * ip_penalty * ui * vi) *
dx;
M12(i, j) += (.5 * nui * dnvi * ue - .5 * nue * dnue * vi -
nu * ip_penalty * vi * ue) *
dx;
M21(i, j) += (-.5 * nue * dnve * ui + .5 * nui * dnui * ve -
nu * ip_penalty * ui * ve) *
dx;
M22(i, j) += (.5 * nue * dnve * ue + .5 * nue * dnue * ve +
nu * ip_penalty * ue * ve) *
dx;
}
}
}
}
}
namespace RHSIntegrator
{
template <int dim>
{}
template <int dim>
{
const double penalty = 2. * degree * (degree + 1) *
dinfo.
face->measure() / dinfo.
cell->measure();
local_vector(i) +=
+
* boundary_values[k] * fe.
JxW(k);
}
template <int dim>
{}
}
namespace Estimator
{
template <int dim>
{
const std::vector<Tensor<2, dim>> &DDuh = info.
hessians[0][0];
{
const double t = dinfo.
cell->diameter() *
trace(DDuh[k]);
}
}
template <int dim>
{
const std::vector<double> &uh = info.
values[0][0];
const double penalty = 2. * degree * (degree + 1) *
dinfo.
face->measure() / dinfo.
cell->measure();
{
const double diff = boundary_values[k] - uh[k];
dinfo.
value(0) += penalty * diff * diff * fe.
JxW(k);
}
}
template <int dim>
{
const std::vector<double> &uh1 = info1.
values[0][0];
const std::vector<double> &uh2 = info2.
values[0][0];
const std::vector<Tensor<1, dim>> &Duh1 = info1.
gradients[0][0];
const std::vector<Tensor<1, dim>> &Duh2 = info2.
gradients[0][0];
const double penalty1 =
degree * (degree + 1) * dinfo1.
face->measure() / dinfo1.
cell->measure();
const double penalty2 =
degree * (degree + 1) * dinfo2.
face->measure() / dinfo2.
cell->measure();
const double penalty = penalty1 + penalty2;
const double h = dinfo1.
face->measure();
{
const double diff1 = uh1[k] - uh2[k];
const double diff2 =
(penalty * diff1 * diff1 + h * diff2 * diff2) * fe.
JxW(k);
}
}
}
namespace ErrorIntegrator
{
template <int dim>
{
const std::vector<Tensor<1, dim>> &Duh = info.
gradients[0][0];
const std::vector<double> &uh = info.
values[0][0];
{
for (
unsigned int d = 0;
d < dim; ++
d)
{
const double diff = exact_gradients[k][
d] - Duh[k][
d];
}
const double diff = exact_values[k] - uh[k];
dinfo.
value(1) += diff * diff * fe.
JxW(k);
}
}
template <int dim>
{
const std::vector<double> &uh = info.
values[0][0];
const double penalty = 2. * degree * (degree + 1) *
dinfo.
face->measure() / dinfo.
cell->measure();
{
const double diff = exact_values[k] - uh[k];
dinfo.
value(0) += penalty * diff * diff * fe.
JxW(k);
}
}
template <int dim>
{
const std::vector<double> &uh1 = info1.
values[0][0];
const std::vector<double> &uh2 = info2.
values[0][0];
const double penalty1 =
degree * (degree + 1) * dinfo1.
face->measure() / dinfo1.
cell->measure();
const double penalty2 =
degree * (degree + 1) * dinfo2.
face->measure() / dinfo2.
cell->measure();
const double penalty = penalty1 + penalty2;
{
const double diff = uh1[k] - uh2[k];
dinfo1.
value(0) += (penalty * diff * diff) * fe.
JxW(k);
}
}
}
template <int dim>
class InteriorPenaltyProblem
{
public:
InteriorPenaltyProblem();
void run(
unsigned int n_steps);
private:
void setup_system();
void assemble_matrix();
void assemble_mg_matrix();
void assemble_right_hand_side();
void error();
double estimate();
void solve();
void output_results(const unsigned int cycle) const;
};
template <int dim>
InteriorPenaltyProblem<dim>::InteriorPenaltyProblem()
, mapping()
, fe(3)
, estimates(1)
{
}
template <int dim>
void InteriorPenaltyProblem<dim>::setup_system()
{
dof_handler.distribute_dofs(fe);
dof_handler.distribute_mg_dofs();
unsigned int n_dofs = dof_handler.n_dofs();
solution.reinit(n_dofs);
right_hand_side.reinit(n_dofs);
sparsity.copy_from(dsp);
mg_matrix.resize(0, n_levels - 1);
mg_matrix.clear_elements();
mg_matrix_dg_up.resize(0, n_levels - 1);
mg_matrix_dg_up.clear_elements();
mg_matrix_dg_down.resize(0, n_levels - 1);
mg_matrix_dg_down.clear_elements();
mg_sparsity.resize(0, n_levels - 1);
mg_sparsity_dg_interface.resize(0, n_levels - 1);
for (
unsigned int level = mg_sparsity.min_level();
level <= mg_sparsity.max_level();
{
mg_sparsity[
level].copy_from(dsp);
{
dof_handler.n_dofs(
level));
mg_sparsity_dg_interface[
level].copy_from(dsp);
mg_matrix_dg_up[
level].reinit(mg_sparsity_dg_interface[
level]);
mg_matrix_dg_down[
level].reinit(mg_sparsity_dg_interface[
level]);
}
}
}
template <int dim>
void InteriorPenaltyProblem<dim>::assemble_matrix()
{
dof_handler.end(),
dof_info,
info_box,
&MatrixIntegrator::cell<dim>,
&MatrixIntegrator::boundary<dim>,
&MatrixIntegrator::face<dim>,
assembler);
}
template <int dim>
void InteriorPenaltyProblem<dim>::assemble_mg_matrix()
{
dof_handler.end_mg(),
dof_info,
info_box,
&MatrixIntegrator::cell<dim>,
&MatrixIntegrator::boundary<dim>,
&MatrixIntegrator::face<dim>,
assembler);
}
template <int dim>
void InteriorPenaltyProblem<dim>::assemble_right_hand_side()
{
dof_handler.end(),
dof_info,
info_box,
&RHSIntegrator::cell<dim>,
&RHSIntegrator::boundary<dim>,
&RHSIntegrator::face<dim>,
assembler);
right_hand_side *= -1.;
}
template <int dim>
void InteriorPenaltyProblem<dim>::solve()
{
mg_transfer.
build(dof_handler);
RELAXATION::AdditionalData smoother_data(1.);
mgmatrix, mg_coarse, mg_transfer, mg_smoother, mg_smoother);
mg.set_edge_flux_matrices(mgdown, mgup);
preconditioner(dof_handler,
mg, mg_transfer);
solver.solve(matrix, solution, right_hand_side, preconditioner);
}
template <int dim>
double InteriorPenaltyProblem<dim>::estimate()
{
std::vector<unsigned int> old_user_indices;
unsigned int i = 0;
cell->set_user_index(i++);
const unsigned int n_gauss_points =
dof_handler.get_fe().tensor_degree() + 1;
n_gauss_points + 1,
n_gauss_points);
info_box.
initialize(fe, mapping, solution_data, solution);
dof_handler.end(),
dof_info,
info_box,
&Estimator::cell<dim>,
&Estimator::boundary<dim>,
&Estimator::face<dim>,
assembler);
return estimates.
block(0).l2_norm();
}
template <int dim>
void InteriorPenaltyProblem<dim>::error()
{
std::vector<unsigned int> old_user_indices;
unsigned int i = 0;
cell->set_user_index(i++);
const unsigned int n_gauss_points =
dof_handler.get_fe().tensor_degree() + 1;
n_gauss_points + 1,
n_gauss_points);
info_box.
initialize(fe, mapping, solution_data, solution);
dof_handler.end(),
dof_info,
info_box,
&ErrorIntegrator::cell<dim>,
&ErrorIntegrator::boundary<dim>,
&ErrorIntegrator::face<dim>,
assembler);
deallog <<
"energy-error: " << errors.
block(0).l2_norm() << std::endl;
deallog <<
"L2-error: " << errors.
block(1).l2_norm() << std::endl;
}
template <int dim>
void
InteriorPenaltyProblem<dim>::output_results(const unsigned int cycle) const
{
const std::string filename =
deallog <<
"Writing solution to <" << filename <<
">..." << std::endl
<< std::endl;
std::ofstream gnuplot_output(filename);
}
template <int dim>
void InteriorPenaltyProblem<dim>::run(unsigned int n_steps)
{
deallog <<
"Element: " << fe.get_name() << std::endl;
for (unsigned int s = 0; s < n_steps; ++s)
{
deallog <<
"Step " << s << std::endl;
if (estimates.
block(0).size() == 0)
else
{
}
<< std::endl;
setup_system();
deallog <<
"DoFHandler " << dof_handler.n_dofs() <<
" dofs, level dofs";
deallog <<
' ' << dof_handler.n_dofs(l);
deallog <<
"Assemble matrix" << std::endl;
assemble_matrix();
deallog <<
"Assemble multilevel matrix" << std::endl;
assemble_mg_matrix();
deallog <<
"Assemble right hand side" << std::endl;
assemble_right_hand_side();
solve();
error();
deallog <<
"Estimate " << estimate() << std::endl;
output_results(s);
}
}
}
int main()
{
try
{
using namespace Step39;
std::ofstream logfile("deallog");
InteriorPenaltyProblem<2> test1;
test1.run(12);
}
catch (std::exception &exc)
{
std::cerr << std::endl
<< std::endl
<< "----------------------------------------------------"
<< std::endl;
std::cerr << "Exception on processing: " << std::endl
<< exc.what() << std::endl
<< "Aborting!" << std::endl
<< "----------------------------------------------------"
<< std::endl;
return 1;
}
catch (...)
{
std::cerr << std::endl
<< std::endl
<< "----------------------------------------------------"
<< std::endl;
std::cerr << "Unknown exception!" << std::endl
<< "Aborting!" << std::endl
<< "----------------------------------------------------"
<< std::endl;
return 1;
}
return 0;
}
void write_gnuplot(std::ostream &out) const
void add_data_vector(const VectorType &data, const std::vector< std::string > &names, const DataVectorType type=type_automatic, const std::vector< DataComponentInterpretation::DataComponentInterpretation > &data_component_interpretation={})
virtual void build_patches(const unsigned int n_subdivisions=0)
const std::vector< Point< spacedim > > & get_quadrature_points() const
const unsigned int dofs_per_cell
const Tensor< 1, spacedim > & normal_vector(const unsigned int q_point) const
const unsigned int n_quadrature_points
const Tensor< 1, spacedim > & shape_grad(const unsigned int i, const unsigned int q_point) const
double JxW(const unsigned int q_point) const
const double & shape_value(const unsigned int i, const unsigned int q_point) const
virtual void gradient_list(const std::vector< Point< dim > > &points, std::vector< Tensor< 1, dim > > &gradients, const unsigned int component=0) const override
virtual void value_list(const std::vector< Point< dim > > &points, std::vector< double > &values, const unsigned int component=0) const override
void set_steps(const unsigned int)
void set_symmetric(const bool)
void set_variable(const bool)
void initialize(AnyData &results, bool separate_faces=true)
void initialize_fluxes(MGLevelObject< MatrixType > &flux_up, MGLevelObject< MatrixType > &flux_down)
void initialize(AnyData &results)
void initialize(const FiniteElement< dim, spacedim > &el, const Mapping< dim, spacedim > &mapping, const BlockInfo *block_info=nullptr)
MeshWorker::VectorSelector boundary_selector
void add_update_flags_cell(const UpdateFlags flags)
MeshWorker::VectorSelector face_selector
MeshWorker::VectorSelector cell_selector
void add_update_flags_boundary(const UpdateFlags flags)
std::vector< std::vector< std::vector< Tensor< 2, spacedim > > > > hessians
const FEValuesBase< dim, spacedim > & fe_values() const
Access to finite element.
std::vector< std::vector< std::vector< Tensor< 1, spacedim > > > > gradients
std::vector< std::vector< std::vector< double > > > values
void add(const std::string &name, const bool values=true, const bool gradients=false, const bool hessians=false)
void initialize(const MGLevelObject< MatrixType2 > &matrices, const typename RelaxationType::AdditionalData &additional_data=typename RelaxationType::AdditionalData())
@ matrix
Contents is actually a matrix.
Tensor< 2, dim, Number > l(const Tensor< 2, dim, Number > &F, const Tensor< 2, dim, Number > &dF_dt)
SymmetricTensor< 2, dim, Number > d(const Tensor< 2, dim, Number > &F, const Tensor< 2, dim, Number > &dF_dt)
T sum(const T &t, const MPI_Comm mpi_communicator)
void run(const Iterator &begin, const std_cxx20::type_identity_t< Iterator > &end, Worker worker, Copier copier, const ScratchData &sample_scratch_data, const CopyData &sample_copy_data, const unsigned int queue_length, const unsigned int chunk_size)
1.11.0