ExactErrorEstimator Class Reference

#include <exact_error_estimator.h>

Inheritance diagram for ExactErrorEstimator:

[legend]

List of all members.

Public Types
typedef std::map< std::pair < const System *, unsigned int > , ErrorVector * >	ErrorMap
Public Member Functions
	ExactErrorEstimator ()
	~ExactErrorEstimator ()
void	attach_exact_value (Number fptr(const Point &p, const Parameters &Parameters, const std::string &sys_name, const std::string &unknown_name))
void	attach_exact_deriv (Gradient fptr(const Point &p, const Parameters &parameters, const std::string &sys_name, const std::string &unknown_name))
void	attach_exact_hessian (Tensor fptr(const Point &p, const Parameters &parameters, const std::string &sys_name, const std::string &unknown_name))
void	attach_reference_solution (EquationSystems *es_fine)
void	extra_quadrature_order (const int extraorder)
virtual void	estimate_error (const System &system, ErrorVector &error_per_cell, const NumericVector< Number > *solution_vector=NULL, bool estimate_parent_error=false)
virtual void	estimate_errors (const EquationSystems &equation_systems, ErrorVector &error_per_cell, const std::map< const System *, SystemNorm > &error_norms, const std::map< const System *, const NumericVector< Number > * > *solution_vectors=NULL, bool estimate_parent_error=false)
virtual void	estimate_errors (const EquationSystems &equation_systems, ErrorMap &errors_per_cell, const std::map< const System *, const NumericVector< Number > * > *solution_vectors=NULL, bool estimate_parent_error=false)
Public Attributes
SystemNorm	error_norm
Protected Member Functions
void	reduce_error (std::vector< float > &error_per_cell) const
Private Member Functions
Real	find_squared_element_error (const System &system, const std::string &var_name, const Elem *elem, const DenseVector< Number > &Uelem, FEBase *fe, MeshFunction *fine_values) const
Private Attributes
Number(*	_exact_value )(const Point &p, const Parameters &parameters, const std::string &sys_name, const std::string &unknown_name)
Gradient(*	_exact_deriv )(const Point &p, const Parameters &parameters, const std::string &sys_name, const std::string &unknown_name)
Tensor(*	_exact_hessian )(const Point &p, const Parameters &parameters, const std::string &sys_name, const std::string &unknown_name)
EquationSystems *	_equation_systems_fine
int	_extra_order

---

Detailed Description

This class implements an "error estimator" based on the difference between the approximate and exact solution. In theory the quadrature error in this estimate should be much lower than the approximation error in other estimates, so this estimator can be used to calculate effectivity.

Author:

Roy Stogner, 2006.

Definition at line 62 of file exact_error_estimator.h.

---

Member Typedef Documentation

typedef std::map<std::pair<const System*, unsigned int>, ErrorVector*> ErrorEstimator::ErrorMap [inherited]

When calculating many error vectors at once, we need a data structure to hold them all

Definition at line 105 of file error_estimator.h.

---

Constructor & Destructor Documentation

ExactErrorEstimator::ExactErrorEstimator

(

)

[inline]

Constructor. Responsible for initializing the _bc_function function pointer to NULL, and defaulting the norm type to H1.

Definition at line 70 of file exact_error_estimator.h.

References ErrorEstimator::error_norm, and libMeshEnums::H1.

                        : _exact_value(NULL), 
                          _exact_deriv(NULL),
                          _exact_hessian(NULL),
                          _extra_order(0)
  { error_norm = H1; }

ExactErrorEstimator::~ExactErrorEstimator

(

)

[inline]

Destructor.

Definition at line 79 of file exact_error_estimator.h.

{}

---

Member Function Documentation

void ExactErrorEstimator::attach_exact_deriv

(

Gradient

fptrconst Point &p,const Parameters &parameters,const std::string &sys_name,const std::string &unknown_name

)

Attach function similar to system.h which allows the user to attach an arbitrary function which computes the exact derivative of the solution at any point.

Definition at line 58 of file exact_error_estimator.C.

References _equation_systems_fine, and _exact_deriv.

{
  libmesh_assert (fptr != NULL);
  _exact_deriv = fptr;

  // If we're not using a fine grid solution
  _equation_systems_fine = NULL;

}

void ExactErrorEstimator::attach_exact_hessian

(

Tensor

fptrconst Point &p,const Parameters &parameters,const std::string &sys_name,const std::string &unknown_name

)

Attach function similar to system.h which allows the user to attach an arbitrary function which computes the exact second derivatives of the solution at any point.

Definition at line 72 of file exact_error_estimator.C.

References _equation_systems_fine, and _exact_hessian.

{
  libmesh_assert (fptr != NULL);
  _exact_hessian = fptr;

  // If we're not using a fine grid solution
  _equation_systems_fine = NULL;
}

void ExactErrorEstimator::attach_exact_value

(

Number

fptrconst Point &p, const Parameters &Parameters, const std::string &sys_name, const std::string &unknown_name

)

Attach function similar to system.h which allows the user to attach an arbitrary function which computes the exact value of the solution at any point.

void ExactErrorEstimator::attach_reference_solution

(

EquationSystems *

es_fine

)

Attach function similar to system.h which allows the user to attach a second EquationSystems object with a reference fine grid solution.

Definition at line 84 of file exact_error_estimator.C.

References _equation_systems_fine, _exact_deriv, _exact_hessian, and _exact_value.

{
  libmesh_assert (es_fine != NULL);
  _equation_systems_fine = es_fine;

  // If we're using a fine grid solution, we're not using exact values
  _exact_value = NULL;
  _exact_deriv = NULL;
  _exact_hessian = NULL;
}

void ExactErrorEstimator::estimate_error	(	const System &	system,
		ErrorVector &	error_per_cell,
		const NumericVector< Number > *	solution_vector = NULL,
		bool	estimate_parent_error = false
	)			[virtual]

This function uses the exact solution function to estimate the error on each cell. The estimated error is output in the vector error_per_cell

Implements ErrorEstimator.

Definition at line 96 of file exact_error_estimator.C.

References Elem::active(), MeshBase::active_local_elements_begin(), MeshBase::active_local_elements_end(), NumericVector< T >::build(), FEBase::build(), Elem::child(), FEBase::coarsened_dof_values(), System::current_local_solution, System::current_solution(), FEType::default_quadrature_rule(), dim, DofMap::dof_indices(), AutoPtr< Tp >::get(), System::get_dof_map(), System::get_mesh(), DofObject::id(), MeshBase::max_elem_id(), mesh, MeshBase::mesh_dimension(), Elem::n_children(), System::n_vars(), n_vars, System::name(), Elem::parent(), libMeshEnums::SERIAL, System::solution, NumericVector< T >::swap(), System::update_global_solution(), System::variable_name(), System::variable_number(), and DofMap::variable_type().

{
  // The current mesh
  const MeshBase& mesh = system.get_mesh();

  // The dimensionality of the mesh
  const unsigned int dim = mesh.mesh_dimension();
  
  // The number of variables in the system
  const unsigned int n_vars = system.n_vars();
  
  // The DofMap for this system
  const DofMap& dof_map = system.get_dof_map();

  // Resize the error_per_cell vector to be
  // the number of elements, initialize it to 0.
  error_per_cell.resize (mesh.max_elem_id());
  std::fill (error_per_cell.begin(), error_per_cell.end(), 0.);

  // Prepare current_local_solution to localize a non-standard
  // solution vector if necessary
  if (solution_vector && solution_vector != system.solution.get())
    {
      NumericVector<Number>* newsol =
        const_cast<NumericVector<Number>*>(solution_vector);
      System &sys = const_cast<System&>(system);
      newsol->swap(*sys.solution);
      sys.update();
    }
  
  // Loop over all the variables in the system
  for (unsigned int var=0; var<n_vars; var++)
    {
      // Possibly skip this variable
      if (error_norm.weight(var) == 0.0) continue;

      // The (string) name of this variable
      const std::string& var_name = system.variable_name(var);
      
      // The type of finite element to use for this variable
      const FEType& fe_type = dof_map.variable_type (var);
      
      AutoPtr<FEBase> fe (FEBase::build (dim, fe_type));

      // Build an appropriate Gaussian quadrature rule
      AutoPtr<QBase> qrule =
        fe_type.default_quadrature_rule (dim,
                                         _extra_order);

      fe->attach_quadrature_rule (qrule.get());

      // Prepare a global solution and a MeshFunction of the fine system if we need one
      AutoPtr<MeshFunction> fine_values;
      AutoPtr<NumericVector<Number> > fine_soln = NumericVector<Number>::build();
      if (_equation_systems_fine)
      {
        const System& fine_system = _equation_systems_fine->get_system(system.name());

        std::vector<Number> global_soln;
        // FIXME - we're assuming that the fine system solution gets
        // used even when a different vector is used for the coarse
        // system
        fine_system.update_global_solution(global_soln);
        fine_soln->init (global_soln.size(), true, SERIAL);
        (*fine_soln) = global_soln;

        fine_values = AutoPtr<MeshFunction>
          (new MeshFunction(*_equation_systems_fine,
                            *fine_soln,
                            fine_system.get_dof_map(),
                            fine_system.variable_number(var_name)));
        fine_values->init();
      }

      // Request the data we'll need to compute with
      fe->get_JxW();
      fe->get_phi();
      fe->get_dphi();
#ifdef LIBMESH_ENABLE_SECOND_DERIVATIVES
      fe->get_d2phi();
#endif
      fe->get_xyz();

      // Iterate over all the active elements in the mesh
      // that live on this processor.
      MeshBase::const_element_iterator
        elem_it  = mesh.active_local_elements_begin();
      const MeshBase::const_element_iterator
        elem_end = mesh.active_local_elements_end(); 

      for (;elem_it != elem_end; ++elem_it)
        {
          // e is necessarily an active element on the local processor
          const Elem* elem = *elem_it;
          const unsigned int e_id = elem->id();

#ifdef LIBMESH_ENABLE_AMR
          // See if the parent of element e has been examined yet;
          // if not, we may want to compute the estimator on it
          const Elem* parent = elem->parent();

          // We only can compute and only need to compute on
          // parents with all active children
          bool compute_on_parent = true;
          if (!parent || !estimate_parent_error)
            compute_on_parent = false;
          else
            for (unsigned int c=0; c != parent->n_children(); ++c)
              if (!parent->child(c)->active())
                compute_on_parent = false;

          if (compute_on_parent &&
              !error_per_cell[parent->id()])
            {
              // Compute a projection onto the parent
              DenseVector<Number> Uparent;
              FEBase::coarsened_dof_values(*(system.current_local_solution),
                                           dof_map, parent, Uparent,
                                           var, false);

              error_per_cell[parent->id()] =
                find_squared_element_error(system, var_name, parent, Uparent,
                                           fe.get(), fine_values.get());
            }
#endif

          // Get the local to global degree of freedom maps
          std::vector<unsigned int> dof_indices;
          dof_map.dof_indices (elem, dof_indices, var);
          DenseVector<Number> Uelem(dof_indices.size());
          for (unsigned int i=0; i != dof_indices.size(); ++i)
            Uelem(i) = system.current_solution(dof_indices[i]);

          error_per_cell[e_id] =
            find_squared_element_error(system, var_name, elem, Uelem,
                                       fe.get(), fine_values.get());

        } // End loop over active local elements
    } // End loop over variables


  
  // Each processor has now computed the error contribuions
  // for its local elements.  We need to sum the vector
  // and then take the square-root of each component.  Note
  // that we only need to sum if we are running on multiple
  // processors, and we only need to take the square-root
  // if the value is nonzero.  There will in general be many
  // zeros for the inactive elements.

  // First sum the vector of estimated error values
  this->reduce_error(error_per_cell);

  // Compute the square-root of each component.
  START_LOG("std::sqrt()", "ExactErrorEstimator");
  for (unsigned int i=0; i<error_per_cell.size(); i++)
    {
      
      if (error_per_cell[i] != 0.)
        {
          libmesh_assert (error_per_cell[i] > 0.);
          error_per_cell[i] = std::sqrt(error_per_cell[i]);
        }
      
      
    }
  STOP_LOG("std::sqrt()", "ExactErrorEstimator");

  // If we used a non-standard solution before, now is the time to fix
  // the current_local_solution
  if (solution_vector && solution_vector != system.solution.get())
    {
      NumericVector<Number>* newsol =
        const_cast<NumericVector<Number>*>(solution_vector);
      System &sys = const_cast<System&>(system);
      newsol->swap(*sys.solution);
      sys.update();
    }
}

void ErrorEstimator::estimate_errors	(	const EquationSystems &	equation_systems,
		ErrorMap &	errors_per_cell,
		const std::map< const System *, const NumericVector< Number > * > *	solution_vectors = NULL,
		bool	estimate_parent_error = false
	)			[virtual, inherited]

This virtual function can be redefined in derived classes, but by default it calls estimate_error repeatedly to calculate the requested error vectors.

Currently this function ignores the error_norm.weight() values because it calculates each variable's error individually, unscaled.

The user selects which errors get computed by filling a map with error vectors: If errors_per_cell[&system][v] exists, it will be filled with the error values in variable v of system

FIXME: This is a default implementation - derived classes should reimplement it for efficiency.

Definition at line 92 of file error_estimator.C.

References ErrorEstimator::error_norm, ErrorEstimator::estimate_error(), EquationSystems::get_system(), EquationSystems::n_systems(), System::n_vars(), n_vars, and SystemNorm::type().

{
  SystemNorm old_error_norm = this->error_norm;

  // Find the requested error values from each system
  for (unsigned int s = 0; s != equation_systems.n_systems(); ++s)
    {
      const System &sys = equation_systems.get_system(s);

      unsigned int n_vars = sys.n_vars();

      for (unsigned int v = 0; v != n_vars; ++v)
        {
          // Only fill in ErrorVectors the user asks for
          if (errors_per_cell.find(std::make_pair(&sys, v)) ==
              errors_per_cell.end())
            continue;

          // Calculate error in only one variable
          std::vector<Real> weights(n_vars, 0.0);
          weights[v] = 1.0;
          this->error_norm =
            SystemNorm(std::vector<FEMNormType>(n_vars, old_error_norm.type(0)),
                       weights);

          const NumericVector<Number>* solution_vector = NULL;
          if (solution_vectors &&
              solution_vectors->find(&sys) != solution_vectors->end())
            solution_vector = solution_vectors->find(&sys)->second;

          this->estimate_error
            (sys, *errors_per_cell[std::make_pair(&sys, v)],
             solution_vector, estimate_parent_error);
        }
    }

  // Restore our old state before returning
  this->error_norm = old_error_norm;
}

void ErrorEstimator::estimate_errors	(	const EquationSystems &	equation_systems,
		ErrorVector &	error_per_cell,
		const std::map< const System *, SystemNorm > &	error_norms,
		const std::map< const System *, const NumericVector< Number > * > *	solution_vectors = NULL,
		bool	estimate_parent_error = false
	)			[virtual, inherited]

This virtual function can be redefined in derived classes, but by default computes the sum of the error_per_cell for each system in the equation_systems.

Currently this function ignores the error_norm member variable, and uses the function argument error_norms instead.

This function is named estimate_errors instead of estimate_error because otherwise C++ can get confused.

Reimplemented in UniformRefinementEstimator.

Definition at line 46 of file error_estimator.C.

References ErrorEstimator::error_norm, ErrorEstimator::estimate_error(), EquationSystems::get_system(), and EquationSystems::n_systems().

{
  SystemNorm old_error_norm = this->error_norm;

  // Sum the error values from each system
  for (unsigned int s = 0; s != equation_systems.n_systems(); ++s)
    {
      ErrorVector system_error_per_cell;
      const System &sys = equation_systems.get_system(s);
      if (error_norms.find(&sys) == error_norms.end())
        this->error_norm = old_error_norm;
      else
        this->error_norm = error_norms.find(&sys)->second;

      const NumericVector<Number>* solution_vector = NULL;
      if (solution_vectors &&
          solution_vectors->find(&sys) != solution_vectors->end())
        solution_vector = solution_vectors->find(&sys)->second;

      this->estimate_error(sys, system_error_per_cell,
                           solution_vector, estimate_parent_error);

      if (s)
        {
          libmesh_assert(error_per_cell.size() == system_error_per_cell.size());
          for (unsigned int i=0; i != error_per_cell.size(); ++i)
            error_per_cell[i] += system_error_per_cell[i];
        }
      else
        error_per_cell = system_error_per_cell;
    }

  // Restore our old state before returning
  this->error_norm = old_error_norm;
}

void ExactErrorEstimator::extra_quadrature_order

(

const int

extraorder

)

[inline]

Increases or decreases the order of the quadrature rule used for numerical integration.

Definition at line 127 of file exact_error_estimator.h.

References _extra_order.

    { _extra_order = extraorder; }

Real ExactErrorEstimator::find_squared_element_error	(	const System &	system,
		const std::string &	var_name,
		const Elem *	elem,
		const DenseVector< Number > &	Uelem,
		FEBase *	fe,
		MeshFunction *	fine_values
	)			const [private]

Helper method for calculating on each element

Definition at line 287 of file exact_error_estimator.C.

References _equation_systems_fine, _exact_deriv, _exact_hessian, _exact_value, ErrorEstimator::error_norm, FEBase::get_d2phi(), FEBase::get_dphi(), System::get_equation_systems(), FEBase::get_JxW(), FEBase::get_phi(), FEBase::get_xyz(), MeshFunction::gradient(), libMeshEnums::H1, libMeshEnums::H1_SEMINORM, libMeshEnums::H2, libMeshEnums::H2_SEMINORM, MeshFunction::hessian(), libMeshEnums::L2, libmesh_norm(), System::name(), EquationSystems::parameters, FEBase::reinit(), DenseVector< T >::size(), TypeTensor< T >::size_sq(), TypeVector< T >::size_sq(), SystemNorm::type(), and System::variable_number().

{
  // The (string) name of this system
  const std::string& sys_name = system.name();

  unsigned int var = system.variable_number(var_name);
  
  const Parameters& parameters = system.get_equation_systems().parameters;

  // reinitialize the element-specific data
  // for the current element
  fe->reinit (elem);

  // Get the data we need to compute with
  const std::vector<Real> &                      JxW          = fe->get_JxW();
  const std::vector<std::vector<Real> >&         phi_values   = fe->get_phi();
  const std::vector<std::vector<RealGradient> >& dphi_values  = fe->get_dphi();
  const std::vector<Point>&                      q_point      = fe->get_xyz();
#ifdef LIBMESH_ENABLE_SECOND_DERIVATIVES
  const std::vector<std::vector<RealTensor> >&   d2phi_values = fe->get_d2phi();
#endif

  // The number of shape functions
  const unsigned int n_sf = Uelem.size();

  // The number of quadrature points
  const unsigned int n_qp = JxW.size();

  Real error_val = 0;

  // Begin the loop over the Quadrature points.
  //
  for (unsigned int qp=0; qp<n_qp; qp++)
    {
      // Real u_h = 0.;
      // RealGradient grad_u_h;

      Number u_h = 0.;

      Gradient grad_u_h;
#ifdef LIBMESH_ENABLE_SECOND_DERIVATIVES
      Tensor grad2_u_h;
#endif

      // Compute solution values at the current
      // quadrature point.  This reqiures a sum
      // over all the shape functions evaluated
      // at the quadrature point.
      for (unsigned int i=0; i<n_sf; i++)
        {
          // Values from current solution.
          u_h      += phi_values[i][qp]*Uelem(i);
          grad_u_h += dphi_values[i][qp]*Uelem(i);
#ifdef LIBMESH_ENABLE_SECOND_DERIVATIVES
          grad2_u_h += d2phi_values[i][qp]*Uelem(i);
#endif
        }

      // Compute the value of the error at this quadrature point
      if (error_norm.type(var) == L2 ||
          error_norm.type(var) == H1 ||
          error_norm.type(var) == H2)
        {
          Number val_error = 0;
          if(_exact_value)
            val_error = u_h - _exact_value(q_point[qp],parameters,sys_name,var_name);
          else if(_equation_systems_fine)
            val_error = u_h - (*fine_values)(q_point[qp]);

          // Add the squares of the error to each contribution
          error_val += JxW[qp]*libmesh_norm(val_error);
        }

      // Compute the value of the error in the gradient at this
      // quadrature point
      if ((_exact_deriv || _equation_systems_fine) && 
          (error_norm.type(var) == H1 ||
           error_norm.type(var) == H1_SEMINORM ||
           error_norm.type(var) == H2))
        {
          Gradient grad_error;
          if(_exact_deriv)
            grad_error = grad_u_h - _exact_deriv(q_point[qp],parameters,sys_name,var_name);
          else if(_equation_systems_fine)
            grad_error = grad_u_h - fine_values->gradient(q_point[qp]);

          error_val += JxW[qp]*grad_error.size_sq();
        }


#ifdef LIBMESH_ENABLE_SECOND_DERIVATIVES
      // Compute the value of the error in the hessian at this
      // quadrature point
      if ((_exact_hessian || _equation_systems_fine) && 
          (error_norm.type(var) == H2_SEMINORM ||
           error_norm.type(var) == H2))
        {
          Tensor grad2_error;
          if(_exact_hessian)
            grad2_error = grad2_u_h - _exact_hessian(q_point[qp],parameters,sys_name,var_name);
          else if (_equation_systems_fine)
            grad2_error = grad2_u_h - fine_values->hessian(q_point[qp]);

          error_val += JxW[qp]*grad2_error.size_sq();
        }
#endif

    } // end qp loop

  libmesh_assert (error_val >= 0.);
          
  return error_val;
}

void ErrorEstimator::reduce_error

(

std::vector< float > &

error_per_cell

)

const [protected, inherited]

This method takes the local error contributions in error_per_cell from each processor and combines them to get the global error vector.

Referenced by PatchRecoveryErrorEstimator::estimate_error(), and JumpErrorEstimator::estimate_error().

---

Member Data Documentation

EquationSystems* ExactErrorEstimator::_equation_systems_fine [private]

Constant pointer to the EquationSystems object containing the fine grid solution.

Definition at line 182 of file exact_error_estimator.h.

Referenced by attach_exact_deriv(), attach_exact_hessian(), attach_reference_solution(), and find_squared_element_error().

Gradient(* ExactErrorEstimator::_exact_deriv)(const Point &p, const Parameters &parameters, const std::string &sys_name, const std::string &unknown_name) [private]

Function pointer to user-provided function which computes the exact derivative of the solution.

Referenced by attach_exact_deriv(), attach_reference_solution(), and find_squared_element_error().

Tensor(* ExactErrorEstimator::_exact_hessian)(const Point &p, const Parameters &parameters, const std::string &sys_name, const std::string &unknown_name) [private]

Function pointer to user-provided function which computes the exact hessian of the solution.

Referenced by attach_exact_hessian(), attach_reference_solution(), and find_squared_element_error().

Number(* ExactErrorEstimator::_exact_value)(const Point &p, const Parameters &parameters, const std::string &sys_name, const std::string &unknown_name) [private]

Function pointer to user-provided function which computes the exact value of the solution.

Referenced by attach_reference_solution(), and find_squared_element_error().

int ExactErrorEstimator::_extra_order [private]

Extra order to use for quadrature rule

Definition at line 197 of file exact_error_estimator.h.

Referenced by extra_quadrature_order().

SystemNorm ErrorEstimator::error_norm [inherited]

When estimating the error in a single system, the error_norm is used to control the scaling and norm choice for each variable. Not all estimators will support all norm choices. The default scaling is for all variables to be weighted equally. The default norm choice depends on the error estimator.

Part of this functionality was supported via component_scale and sobolev_order in older libMesh versions, and a small part was supported via component_mask in even older versions. Hopefully the encapsulation here will allow us to avoid changing this API again.

Definition at line 137 of file error_estimator.h.

Referenced by KellyErrorEstimator::boundary_side_integration(), DiscontinuityMeasure::boundary_side_integration(), DiscontinuityMeasure::DiscontinuityMeasure(), JumpErrorEstimator::estimate_error(), ErrorEstimator::estimate_errors(), ExactErrorEstimator(), find_squared_element_error(), KellyErrorEstimator::internal_side_integration(), LaplacianErrorEstimator::internal_side_integration(), DiscontinuityMeasure::internal_side_integration(), KellyErrorEstimator::KellyErrorEstimator(), LaplacianErrorEstimator::LaplacianErrorEstimator(), PatchRecoveryErrorEstimator::EstimateError::operator()(), PatchRecoveryErrorEstimator::PatchRecoveryErrorEstimator(), and UniformRefinementEstimator::UniformRefinementEstimator().

---

The documentation for this class was generated from the following files:

• exact_error_estimator.h
• exact_error_estimator.C