Reference documentation for deal.II version Git 7a0e96d111 20210621 21:20:26 0400

Typedefs  
template<typename Number >  
using  ComparisonFunction = std::function< bool(const Number &, const Number &)> 
Functions  
Setting padaptivity flags  
template<int dim, int spacedim>  
void  full_p_adaptivity (const ::DoFHandler< dim, spacedim > &dof_handler) 
template<int dim, int spacedim>  
void  p_adaptivity_from_flags (const ::DoFHandler< dim, spacedim > &dof_handler, const std::vector< bool > &p_flags) 
template<int dim, typename Number , int spacedim>  
void  p_adaptivity_from_absolute_threshold (const ::DoFHandler< dim, spacedim > &dof_handler, const Vector< Number > &criteria, const Number p_refine_threshold, const Number p_coarsen_threshold, const ComparisonFunction< typename identity< Number >::type > &compare_refine=std::greater_equal< Number >(), const ComparisonFunction< typename identity< Number >::type > &compare_coarsen=std::less_equal< Number >()) 
template<int dim, typename Number , int spacedim>  
void  p_adaptivity_from_relative_threshold (const ::DoFHandler< dim, spacedim > &dof_handler, const Vector< Number > &criteria, const double p_refine_fraction=0.5, const double p_coarsen_fraction=0.5, const ComparisonFunction< typename identity< Number >::type > &compare_refine=std::greater_equal< Number >(), const ComparisonFunction< typename identity< Number >::type > &compare_coarsen=std::less_equal< Number >()) 
template<int dim, typename Number , int spacedim>  
void  p_adaptivity_fixed_number (const ::DoFHandler< dim, spacedim > &dof_handler, const Vector< Number > &criteria, const double p_refine_fraction=0.5, const double p_coarsen_fraction=0.5, const ComparisonFunction< typename identity< Number >::type > &compare_refine=std::greater_equal< Number >(), const ComparisonFunction< typename identity< Number >::type > &compare_coarsen=std::less_equal< Number >()) 
template<int dim, typename Number , int spacedim>  
void  p_adaptivity_from_regularity (const ::DoFHandler< dim, spacedim > &dof_handler, const Vector< Number > &sobolev_indices) 
template<int dim, typename Number , int spacedim>  
void  p_adaptivity_from_reference (const ::DoFHandler< dim, spacedim > &dof_handler, const Vector< Number > &criteria, const Vector< Number > &references, const ComparisonFunction< typename identity< Number >::type > &compare_refine, const ComparisonFunction< typename identity< Number >::type > &compare_coarsen) 
Error prediction  
template<int dim, typename Number , int spacedim>  
void  predict_error (const ::DoFHandler< dim, spacedim > &dof_handler, const Vector< Number > &error_indicators, Vector< Number > &predicted_errors, const double gamma_p=std::sqrt(0.4), const double gamma_h=2., const double gamma_n=1.) 
Decide between h and padaptivity  
template<int dim, int spacedim>  
void  force_p_over_h (const ::DoFHandler< dim, spacedim > &dof_handler) 
template<int dim, int spacedim>  
void  choose_p_over_h (const ::DoFHandler< dim, spacedim > &dof_handler) 
Optimiize plevel distribution  
template<int dim, int spacedim>  
bool  limit_p_level_difference (const ::DoFHandler< dim, spacedim > &dof_handler, const unsigned int max_difference=1, const unsigned int contains_fe_index=0) 
We supply adaptive methods to align computational resources with the complexity of the numerical solution. Error estimates are an appropriate means of determining where adjustments need to be made.
However with hpadaptivity, we have two ways to realize these adjustments: For irregular solutions, hadaptive methods which dynamically assign cell sizes tend to reduce the approximation error, while for smooth solutions padaptive methods are better suited in which function spaces will be selected dynamically. This namespace collects tools to decide which type of adaptive methods to apply.
To successfully apply hpadaptive methods, we recommend the following workflow:
A suitable error estimate is the basis for any kind of adaptive method. Similar to pure grid refinement, we will determine error estimates in the usual way (i.e. KellyErrorEstimator) and mark cells for refinement or coarsening (i.e. GridRefinement).
Calling Triangulation::execute_coarsening_and_refinement() at this stage will perform pure grid refinement as expected.
Once all refinement and coarsening flags have been distributed on the mesh, we may determine if those qualify for padaptive methods. Corresponding functions will set future_fe_indices
on top of the refinement and coarsening flags if they fulfil a certain criterion.
In case of refinement, the superordinate element of the underlying hp::FECollection will be assigned as the future finite element. Correspondingly, the subordinate element will be selected for coarsening.
Triangulation::execute_coarsening_and_refinement() will now supply both h and padaptive methods independently.
Right now, there may be cells scheduled for both h and padaptation. If we do not want to impose both methods at once, we need to decide which one to pick for each cell individually and unambiguously. Since grid refinement will be imposed by default and we only determine qualification for padaptivity on top, we will always decide in favour of padaptive methods.
Calling Triangulation::execute_coarsening_and_refinement() will now perform either h or padaptive methods uniquely on each cell.
future_fe_indices
will be used to update the data accordingly. As an example, a realisation of pure padaptive methods would look like the following:
using hp::Refinement::ComparisonFunction = typedef std::function<bool(const Number &, const Number &)> 
An alias that defines the characteristics of a function that can be used as a comparison criterion for deciding whether to perform h or padaptation.
Such functions take two numbers as arguments: The first one corresponds to the provided criterion, while the other one conforms to the reference. The result of the comparison will be returned as a boolean.
Definition at line 139 of file refinement.h.
void hp::Refinement::full_p_adaptivity  (  const ::DoFHandler< dim, spacedim > &  dof_handler  ) 
Each cell flagged for hrefinement will also be flagged for prefinement. The same applies to coarsening.
Setting padaptivity flags
Definition at line 47 of file refinement.cc.
void hp::Refinement::p_adaptivity_from_flags  (  const ::DoFHandler< dim, spacedim > &  dof_handler, 
const std::vector< bool > &  p_flags  
) 
Adapt which finite element to use on cells that have been specifically flagged for padaptation via the parameter p_flags
. Future finite elements will only be assigned if cells have been flagged for refinement and coarsening beforehand.
Each entry of the parameter p_flags
needs to correspond to an active cell.
Definition at line 67 of file refinement.cc.
void hp::Refinement::p_adaptivity_from_absolute_threshold  (  const ::DoFHandler< dim, spacedim > &  dof_handler, 
const Vector< Number > &  criteria,  
const Number  p_refine_threshold,  
const Number  p_coarsen_threshold,  
const ComparisonFunction< typename identity< Number >::type > &  compare_refine = std::greater_equal<Number>() , 

const ComparisonFunction< typename identity< Number >::type > &  compare_coarsen = std::less_equal<Number>() 

) 
Adapt which finite element to use on cells whose criteria meet a certain absolute threshold.
For prefinement and pcoarsening, two separate thresholds need to provided via parameters p_refine_threshold
and p_coarsen_threshold
.
We consider a cell for padaptivity if it is currently flagged for refinement or coarsening and its criterion successfully compares to the corresponding threshold. Let us be more specific on the default case: We consider a cell for prefinement if it is flagged for refinement and its criterion is larger than or equal to the corresponding threshold. The same applies for pcoarsening, but the cell's criterion must be lower than or equal to the threshold. However, different compare function objects can be supplied via the parameters compare_refine
and compare_coarsen
to impose different decision strategies.
Each entry of the parameter criteria
needs to correspond to an active cell.
Definition at line 111 of file refinement.cc.
void hp::Refinement::p_adaptivity_from_relative_threshold  (  const ::DoFHandler< dim, spacedim > &  dof_handler, 
const Vector< Number > &  criteria,  
const double  p_refine_fraction = 0.5 , 

const double  p_coarsen_fraction = 0.5 , 

const ComparisonFunction< typename identity< Number >::type > &  compare_refine = std::greater_equal<Number>() , 

const ComparisonFunction< typename identity< Number >::type > &  compare_coarsen = std::less_equal<Number>() 

) 
Adapt which finite element to use on cells whose criteria meet a certain threshold relative to the overall range of criterion values.
The threshold will be determined for refined and coarsened cells separately based on the currently set refinement markers. For each class of cells, we determine the maximal and minimal values of all criteria and determine the threshold by linear interpolation between these limits. Parameters p_refine_fraction
and p_refine_coarsen
are used as interpolation factors, where 0
corresponds to the minimal and 1
to the maximal value. By default, mean values are considered as thresholds.
We consider a cell for padaptivity if it is currently flagged for refinement or coarsening and its criterion successfully compares to the corresponding threshold. Let us be more specific on the default case: We consider a cell for prefinement if it is flagged for refinement and its criterion is larger than or equal to the corresponding threshold. The same applies for pcoarsening, but the cell's criterion must be lower than or equal to the threshold. However, different compare function objects can be supplied via the parameters compare_refine
and compare_coarsen
to impose different decision strategies.
Each entry of the parameter criteria
needs to correspond to an active cell. Parameters p_refine_fraction
and p_coarsen_fraction
need to be in the interval \([0,1]\).
Definition at line 150 of file refinement.cc.
void hp::Refinement::p_adaptivity_fixed_number  (  const ::DoFHandler< dim, spacedim > &  dof_handler, 
const Vector< Number > &  criteria,  
const double  p_refine_fraction = 0.5 , 

const double  p_coarsen_fraction = 0.5 , 

const ComparisonFunction< typename identity< Number >::type > &  compare_refine = std::greater_equal<Number>() , 

const ComparisonFunction< typename identity< Number >::type > &  compare_coarsen = std::less_equal<Number>() 

) 
Adapt which finite element to use on a given fraction of cells.
Out of all cells flagged for a certain type of adaptation, be it refinement or coarsening, we will determine a fixed number of cells among this subset that will be flagged for the corresponding padaptive variant.
For each of both refinement and coarsening subsets, we will determine a threshold based on the provided parameter criteria
containing indicators for every active cell. In the default case for refinement, all cells with an indicator larger than or equal to the corresponding threshold will be considered for prefinement, while for coarsening all cells with an indicator less than or equal to the matching threshold are taken into account. However, different compare function objects can be supplied via the parameters compare_refine
and compare_coarsen
to impose different decision strategies.
For refinement, the threshold will be associated with the cell that has the p_refine_fraction
times Triangulation::n_active_cells() largest indicator, while it is the cell with the p_refine_coarsen
times Triangulation::n_active_cells() lowest indicator for coarsening.
Each entry of the parameter criteria
needs to correspond to an active cell. Parameters p_refine_fraction
and p_coarsen_fraction
need to be in the interval \([0,1]\).
Definition at line 247 of file refinement.cc.
void hp::Refinement::p_adaptivity_from_regularity  (  const ::DoFHandler< dim, spacedim > &  dof_handler, 
const Vector< Number > &  sobolev_indices  
) 
Adapt which finite element to use on cells based on the regularity of the (unknown) analytical solution.
With an approximation of the local Sobolev regularity index \(k_K\), we may assess to which finite element space our local solution on cell \(K\) belongs. Since the regularity index is only an estimate, we won't use it to assign the finite element space directly, but rather consider it as an indicator for adaptation. If a cell is flagged for refinement, we will perform prefinement once it satisfies \(k_K > p_{K,\text{super}}\), where \(p_{K,\text{super}}\) is the polynomial degree of the finite element superordinate to the currently active element on cell \(K\). In case of coarsening, the criterion \(k_K < p_{K,\text{sub}}\) has to be met, with \(p_{K,\text{sub}}\) the degree of the subordinate element.
Each entry of the parameter sobolev_indices
needs to correspond to an active cell.
For more theoretical details see [2] .
Definition at line 448 of file refinement.cc.
void hp::Refinement::p_adaptivity_from_reference  (  const ::DoFHandler< dim, spacedim > &  dof_handler, 
const Vector< Number > &  criteria,  
const Vector< Number > &  references,  
const ComparisonFunction< typename identity< Number >::type > &  compare_refine,  
const ComparisonFunction< typename identity< Number >::type > &  compare_coarsen  
) 
Adapt which finite element to use on each cell based on how its criterion relates to a reference.
We consider a cell for padaptivity if it is currently flagged for refinement or coarsening and its criterion successfully compares to the corresponding reference. Other than functions p_adaptivity_from_absolute_threshold() and p_adaptivity_from_relative_threshold(), compare function objects have to be provided explicitly via the parameters compare_refine
and compare_coarsen
.
Each entry of the parameters criteria
and references
needs to correspond to an active cell.
Definition at line 506 of file refinement.cc.
void hp::Refinement::predict_error  (  const ::DoFHandler< dim, spacedim > &  dof_handler, 
const Vector< Number > &  error_indicators,  
Vector< Number > &  predicted_errors,  
const double  gamma_p = std::sqrt(0.4) , 

const double  gamma_h = 2. , 

const double  gamma_n = 1. 

) 
Predict how the current error_indicators
will adapt after refinement and coarsening were to happen on the provided dof_handler
, and write its results to predicted_errors
. Each entry of error_indicators
and predicted_errors
corresponds to an active cell on the underlying Triangulation, thus each container has to be of size Triangulation::n_active_cells(). The errors are interpreted to be measured in the energy norm; this assumption enters the rate of convergence that is used in the prediction. The \(l_2\)norm of the output argument predicted_errors
corresponds to the predicted global error after adaptation.
For padaptation, the local error is expected to converge exponentially with the polynomial degree of the assigned finite element. Each increase or decrease of the degree will thus change its value by a userdefined control parameter gamma_p
.
For hadaptation, we expect the local error \(\eta_K\) on cell \(K\) to be proportional to \((h_K)^{p_K}\) in the energy norm, where \(h_K\) denotes the cell diameter and \(p_K\) the polynomial degree of the currently assigned finite element on cell \(K\).
During hcoarsening, the finite elements on siblings may be different, and their parent cell will be assigned to their least dominating finite element that belongs to its most general child. Thus, we will always interpolate on an enclosing finite element space. Additionally assuming that the finite elements on the cells to be coarsened are sufficient to represent the solution correctly (e.g. at least quadratic basis functions for a quadratic solution), we are confident to say that the error will not change by sole interpolation on the larger finite element space.
For padaptation, the local error is expected to converge exponentially with the polynomial degree of the assigned finite element. Each increase or decrease of the degree will thus change its value by a userdefined control parameter gamma_p
. The assumption of exponential convergence is only valid if both h and padaptive methods are combined in a sense that they are both utilitzed throughout a mesh, but do not have to be applied both on a cell simultaneously.
The prediction algorithm is formulated as follows with control parameters gamma_p
, gamma_h
and gamma_n
that may be used to influence prediction for each adaptation type individually. The results for each individual cell are stored in the predicted_errors
output argument.
Adaptation type  Prediction formula  

no adaptation  \(\eta_{K,\text{pred}} = \eta_{K} \, \gamma_\text{n}\)  \(\gamma_\text{n} \in (0,\infty)\) 
padaptation  \(\eta_{K,\text{pred}} = \eta_{K} \, \gamma_\text{p}^{(p_{K,\text{future}}  p_K)}\)  \(\gamma_\text{p} \in (0,1)\) 
hprefinement  \(\eta_{K,\text{pred}} = \eta_{K} \, \gamma_\text{h} \, 0.5^{p_{K,\text{future}}} \, \gamma_\text{p}^{(p_{K,\text{future}}  p_{K})}\)  \(\gamma_\text{h} \in (0,\infty)\) 
hpcoarsening  \(\eta_{K,\text{pred}} = \eta_{K} \, (\gamma_\text{h} \, 0.5^{p_{K,\text{future}}})^{1} \, \gamma_\text{p}^{(p_{K,\text{future}}  p_{K})}\) 
On basis of the refinement history, we use the predicted error estimates to decide how cells will be adapted in the next adaptation step. Comparing the predicted error from the previous adaptation step to the error estimates of the current step allows us to justify whether our previous choice of adaptation was justified, and lets us decide how to adapt in the next one.
We thus have to transfer the predicted error from the old to the adapted mesh. When transferring the predicted error to the adapted mesh, make sure to configure your CellDataTransfer object with AdaptationStrategies::Refinement::l2_norm() as a refinement strategy and AdaptationStrategies::Coarsening::l2_norm() as a coarsening strategy. This ensures that the \(l_2\)norm of the predict errors is preserved on both meshes.
In this context, we assume that the local error on a cell to be hrefined will be divided equally on all of its \(n_{K_c}\) children, whereas local errors on siblings will be summed up on the parent cell in case of hcoarsening. This assumption is often not satisfied in practice: For example, if a cell is at a corner singularity, then the one child cell that ends up closest to the singularity will inherit the majority of the remaining error – but this function can not know where the singularity will be, and consequently assumes equal distribution.
Incorporating the transfer from the old to the adapted mesh, the complete error prediction algorithm reads as follows:
Adaptation type  Prediction formula  

no adaptation  \(\eta_{K,\text{pred}} = \eta_{K} \, \gamma_\text{n}\)  \(\gamma_\text{n} \in (0,\infty)\) 
padaptation  \(\eta_{K,\text{pred}} = \eta_{K} \, \gamma_\text{p}^{(p_{K,\text{future}}  p_K)}\)  \(\gamma_\text{p} \in (0,1)\) 
hprefinement  \(\left( \eta_{K_c,\text{pred}} \right)^2 = n_{K_c}^{1} \left( \eta_{K_p} \, \gamma_\text{h} \, 0.5^{p_{K_c,\text{future}}} \, \gamma_\text{p}^{(p_{K_c,\text{future}}  p_{K_p})} \right)^2 \quad \forall K_c \text{ children of } K_p\)  \(\gamma_\text{h} \in (0,\infty)\) 
hpcoarsening  \(\left( \eta_{K_p,\text{pred}} \right)^2 = \sum\limits_{K_c} \left( \eta_{K_c} \, (\gamma_\text{h} \, 0.5^{p_{K_p,\text{future}}})^{1} \, \gamma_\text{p}^{(p_{K_p,\text{future}}  p_{K_c})} \right)^2 \quad \forall K_c \text{ children of } K_p\) 
With these predicted error estimates, we are capable of adapting the finite element on cells based on their refinement history or rather the predicted change of their error estimates.
If a cell is flagged for adaptation, we want to perform padaptation once the associated error indicators \(\eta_{K}\) on cell \(K\) satisfy \(\eta_{K} < \eta_{K,\text{pred}}\), where the subscript \(\text{pred}\) denotes the predicted error. This corresponds to our assumption of smoothness being correct, else hadaptation is applied. We achieve this with the function hp::Refinement::p_adaptivity_from_reference() and a function object std::less<Number>()
for both comparator parameters.
Also with an alternative strategy, we can determine the fractions of cells to be h and padapted among all cells to be adapted. For this, use hp::Refinement::p_adaptivity_fixed_number() with criteria \((\eta_{K,\text{pred}}  \eta_{K})\).
For the very first adaptation step in either case, the user needs to decide whether h or padaptation is supposed to happen. An hstep will be applied with \(\eta_{K,\text{pred}} = 0\), whereas \(\eta_{K,\text{pred}} = \infty\) ensures a pstep. The latter may be realized with std::numeric_limits::infinity()
.
The following code snippet demonstrates how to impose hpadaptivity based on refinement history in an application:
For more theoretical details see [89] , where the default parameters for this function come from as well, i.e. \(\gamma_\text{p}^2 = 0.4\), \(\gamma_\text{h}^2 = 4\), \(\gamma_\text{n}^2 = 1\).
If you are working with parallel::distributed::Triangulation objects, you need to pay special attention. Here, p4est determines the details of grid refinement, and consequently, it yields more reliable and trustworthy results when we determine the predicted errors during the adaptation process. We can do exactly this by attaching this function to the signal Triangulation::Signals::post_p4est_refinement, which is triggered after p4est got refined, but before data is prepared for transfer. Refinement and coarsening flags of the Triangulation object need to be matched with the already refined p4est oracle using internal::parallel::distributed::TemporarilyMatchRefineFlags. Thus, a construct like the following is necessary to correctly predict errors in parallel distributed applications.
The container predicted_errors
then needs to follow the usual parallel::distributed::CellDataTransfer workflow.
Error prediction
Definition at line 549 of file refinement.cc.
void hp::Refinement::force_p_over_h  (  const ::DoFHandler< dim, spacedim > &  dof_handler  ) 
Choose padaptivity over hadaptivity in any case.
Removes all refine and coarsen flags on cells that have a future_fe_index
assigned.
Decide between h and padaptivity
Definition at line 668 of file refinement.cc.
void hp::Refinement::choose_p_over_h  (  const ::DoFHandler< dim, spacedim > &  dof_handler  ) 
Choose padaptivity over hadaptivity whenever it is invoked on all related cells.
In case of refinement, information about finite elements will be inherited. Thus we will prefer prefinement over hrefinement whenever desired, i.e. clear the refine flag and supply a corresponding future_fe_index
.
However for coarsening, we follow a different approach. Flagging a cell for hcoarsening does not ultimately mean that it will be coarsened. Only if a cell and all of its siblings are flagged, they will be merged into their parent cell. If we consider pcoarsening on top, we must decide for all siblings together how they will be coarsened. We distinguish between three different cases:
future_fe_indices
and clear the coarsen flags on all siblings. future_fe_indices
on all siblings. future_fe_indices
and clear the coarsen flags on all siblings. Definition at line 690 of file refinement.cc.
bool hp::Refinement::limit_p_level_difference  (  const ::DoFHandler< dim, spacedim > &  dof_handler, 
const unsigned int  max_difference = 1 , 

const unsigned int  contains_fe_index = 0 

) 
Limit plevel differences between neighboring cells.
Essentially does to future FE indices what Triangulation::prepare_coarsening_and_refinement() does to refinement flags.
In detail, this function limits the level difference of neighboring cells and thus smoothes the overall function space. Future FE indices will be raised (and never lowered) so that the level difference to neighboring cells is never larger than max_difference
.
Multiple FE hierarchies might have been registered via hp::FECollection::set_hierarchy(). This function operates on only one hierarchy, namely the one that contains the FE index contains_fe_index
. Cells with future FE indices that are not part of the corresponding hierarchy will be ignored.
The function can optionally be called before performing adaptation with Triangulation::execute_coarsening_and_refinement(). It is not necessary to call this function, nor will it be automatically invoked in any part of the library (contrary to its Triangulation counterpart).
On cells that will be hcoarsened, we enforce the difference criterion as if it is already a parent cell. That means, we set the level of all siblings to the highest one among them. In that case, all sibling cells need to have the hcoarsenening flags set terminally via Triangulation::prepare_coarsening_and_refinement() beforehand. Otherwise an assertion will be triggered.
Returns whether any future FE indices have been changed by this function.
Optimize plevel distribution
Definition at line 804 of file refinement.cc.