Items related to the Schwarz linear solver. More...
Namespaces  
Actions  
Actions related to the Schwarz solver.  
OptionTags  
Option tags related to the Schwarz solver.  
protocols  
Protocols related to the Schwarz solver.  
Tags  
Tags related to the Schwarz solver.  
Classes  
struct  ElementCenteredSubdomainData 
Data on an elementcentered subdomain. More...  
class  OverlapIterator 
Iterate over grid points in a region that extends partially into the volume. More...  
struct  Schwarz 
An additive Schwarz subdomain solver for linear systems of equations \(Ax=b\). More...  
Typedefs  
template<size_t Dim>  
using  OverlapId = std::pair< Direction< Dim >, ElementId< Dim > > 
Identifies a subdomain region that overlaps with another element.  
template<size_t Dim, typename ValueType >  
using  OverlapMap = FixedHashMap< maximum_number_of_neighbors(Dim), OverlapId< Dim >, ValueType, boost::hash< OverlapId< Dim > >> 
Data structure that can store the ValueType on each possible overlap of an elementcentered subdomain with its neighbors. Overlaps are identified by their OverlapId .  
Functions  
template<size_t Dim, typename LhsTagsList , typename RhsTagsList >  
decltype(auto)  operator+ (ElementCenteredSubdomainData< Dim, LhsTagsList > lhs, const ElementCenteredSubdomainData< Dim, RhsTagsList > &rhs) noexcept 
template<size_t Dim, typename LhsTagsList , typename RhsTagsList >  
decltype(auto)  operator (ElementCenteredSubdomainData< Dim, LhsTagsList > lhs, const ElementCenteredSubdomainData< Dim, RhsTagsList > &rhs) noexcept 
template<size_t Dim, typename TagsList >  
decltype(auto)  operator* (const double scalar, ElementCenteredSubdomainData< Dim, TagsList > data) noexcept 
template<size_t Dim, typename TagsList >  
decltype(auto)  operator* (ElementCenteredSubdomainData< Dim, TagsList > data, const double scalar) noexcept 
template<size_t Dim, typename TagsList >  
decltype(auto)  operator/ (ElementCenteredSubdomainData< Dim, TagsList > data, const double scalar) noexcept 
template<size_t Dim, typename TagsList >  
std::ostream &  operator<< (std::ostream &os, const ElementCenteredSubdomainData< Dim, TagsList > &subdomain_data) noexcept 
template<size_t Dim, typename TagsList >  
bool  operator== (const ElementCenteredSubdomainData< Dim, TagsList > &lhs, const ElementCenteredSubdomainData< Dim, TagsList > &rhs) noexcept 
template<size_t Dim, typename TagsList >  
bool  operator!= (const ElementCenteredSubdomainData< Dim, TagsList > &lhs, const ElementCenteredSubdomainData< Dim, TagsList > &rhs) noexcept 
size_t  overlap_extent (size_t volume_extent, size_t max_overlap) noexcept 
The number of points that an overlap extends into the volume_extent More...  
template<size_t Dim>  
size_t  overlap_num_points (const Index< Dim > &volume_extents, size_t overlap_extent, size_t overlap_dimension) noexcept 
Total number of grid points in an overlap region that extends overlap_extent points into the volume_extents from either side in the overlap_dimension More...  
double  overlap_width (size_t overlap_extent, const DataVector &collocation_points) noexcept 
Width of an overlap extending overlap_extent points into the collocation_points from either side. More...  
template<size_t Dim, typename DataType , typename... TensorStructure>  
void  data_on_overlap (const gsl::not_null< Tensor< DataType, TensorStructure... > * > restricted_tensor, const Tensor< DataType, TensorStructure... > &tensor, const Index< Dim > &volume_extents, const size_t overlap_extent, const Direction< Dim > &direction) noexcept 
The part of the tensor data that lies within the overlap region.  
template<size_t Dim, typename DataType , typename... TensorStructure>  
Tensor< DataType, TensorStructure... >  data_on_overlap (const Tensor< DataType, TensorStructure... > &tensor, const Index< Dim > &volume_extents, const size_t overlap_extent, const Direction< Dim > &direction) noexcept 
template<size_t Dim, typename OverlapTagsList , typename VolumeTagsList >  
void  data_on_overlap (const gsl::not_null< Variables< OverlapTagsList > * > overlap_data, const Variables< VolumeTagsList > &volume_data, const Index< Dim > &volume_extents, const size_t overlap_extent, const Direction< Dim > &direction) noexcept 
template<size_t Dim, typename TagsList >  
Variables< TagsList >  data_on_overlap (const Variables< TagsList > &volume_data, const Index< Dim > &volume_extents, const size_t overlap_extent, const Direction< Dim > &direction) noexcept 
template<size_t Dim, typename VolumeTagsList , typename OverlapTagsList >  
void  add_overlap_data (const gsl::not_null< Variables< VolumeTagsList > * > volume_data, const Variables< OverlapTagsList > &overlap_data, const Index< Dim > &volume_extents, const size_t overlap_extent, const Direction< Dim > &direction) noexcept 
Add the overlap_data to the volume_data  
template<size_t Dim, typename ExtendedTagsList , typename OverlapTagsList >  
void  extended_overlap_data (const gsl::not_null< Variables< ExtendedTagsList > * > extended_data, const Variables< OverlapTagsList > &overlap_data, const Index< Dim > &volume_extents, const size_t overlap_extent, const Direction< Dim > &direction) noexcept 
Extend the overlap data to the full mesh by filling it with zeros outside the overlap region.  
template<size_t Dim, typename TagsList >  
Variables< TagsList >  extended_overlap_data (const Variables< TagsList > &overlap_data, const Index< Dim > &volume_extents, const size_t overlap_extent, const Direction< Dim > &direction) noexcept 
DataVector  extruding_weight (const DataVector &logical_coords, double width, const Side &side) noexcept 
Weights for the solution on an elementcentered subdomain, decreasing from 1 to 0.5 towards the side over the logical distance width , and further to 0 over the same distance outside the element. More...  
template<size_t Dim>  
void  element_weight (const gsl::not_null< Scalar< DataVector > * > weight, const tnsr::I< DataVector, Dim, Frame::Logical > &logical_coords, const std::array< double, Dim > &overlap_widths, const std::unordered_set< Direction< Dim >> &external_boundaries) noexcept 
template<size_t Dim>  
Scalar< DataVector >  element_weight (const tnsr::I< DataVector, Dim, Frame::Logical > &logical_coords, const std::array< double, Dim > &overlap_widths, const std::unordered_set< Direction< Dim >> &external_boundaries) noexcept 
DataVector  intruding_weight (const DataVector &logical_coords, double width, const Side &side) noexcept 
Weights for the intruding solution of a neighboring elementcentered subdomain, increasing from 0 to 0.5 towards the side over the logical distance width , and further to 1 over the same distance outside the element. More...  
template<size_t Dim>  
void  intruding_weight (const gsl::not_null< Scalar< DataVector > * > weight, const tnsr::I< DataVector, Dim, Frame::Logical > &logical_coords, const Direction< Dim > &direction, const std::array< double, Dim > &overlap_widths, const size_t num_intruding_overlaps, const std::unordered_set< Direction< Dim >> &external_boundaries) noexcept 
template<size_t Dim>  
Scalar< DataVector >  intruding_weight (const tnsr::I< DataVector, Dim, Frame::Logical > &logical_coords, const Direction< Dim > &direction, const std::array< double, Dim > &overlap_widths, const size_t num_intruding_overlaps, const std::unordered_set< Direction< Dim >> &external_boundaries) noexcept 
template<size_t Dim>  
void  element_weight (gsl::not_null< Scalar< DataVector > * > element_weight, const tnsr::I< DataVector, Dim, Frame::Logical > &logical_coords, const std::array< double, Dim > &overlap_widths, const std::unordered_set< Direction< Dim >> &external_boundaries) noexcept 
Weights for data on the central element of an elementcentered subdomain. More...  
template<size_t Dim>  
void  intruding_weight (gsl::not_null< Scalar< DataVector > * > weight, const tnsr::I< DataVector, Dim, Frame::Logical > &logical_coords, const Direction< Dim > &direction, const std::array< double, Dim > &overlap_widths, size_t num_intruding_overlaps, const std::unordered_set< Direction< Dim >> &external_boundaries) noexcept 
Weights for data on overlap regions intruding into an elementcentered subdomain. More...  
Items related to the Schwarz linear solver.
LinearSolver::Schwarz::Schwarz

noexcept 
Weights for data on the central element of an elementcentered subdomain.
Constructs the weighting field
\begin{equation} W(\boldsymbol{\xi}) = \prod^d_{i=0} w(\xi^i) \end{equation}
where \(w(\xi^i)\) is the onedimensional weighting function described in LinearSolver::Schwarz::extruding_weight
and \(\xi^i\) are the elementlogical coordinates (see Eq. (41) in [72]).

noexcept 
Weights for the solution on an elementcentered subdomain, decreasing from 1 to 0.5 towards the side
over the logical distance width
, and further to 0 over the same distance outside the element.
The weighting function over a full elementcentered subdomain is
\begin{equation} w(\xi) = \frac{1}{2}\left( \phi\left( \frac{\xi + 1}{\delta} \right)  \phi\left( \frac{\xi  1}{\delta} \right) \right) \end{equation}
where \(\phi(\xi)\) is a secondorder smoothstep
, i.e. the quintic polynomial
\begin{align*} \phi(\xi) = \begin{cases} \mathrm{sign}(\xi) \quad \text{for} \quad \xi > 1 \\ \frac{1}{8}\left(15\xi  10\xi^3 + 3\xi^5\right) \end{cases} \end{align*}
(see Eq. (39) in [72]).
The LinearSolver::Schwarz::extruding_weight
and LinearSolver::Schwarz::intruding_weight
functions each compute one of the two terms in \(w(\xi)\). For example, consider an elementcentered subdomain A
that overlaps with a neighboring elementcentered subdomain B
. To combine solutions on A
and B
to a weighted solution on A
, multiply the solution on A
with the extruding_weight
and the solution on B
with the intruding_weight
, both evaluated at the logical coordinates in A
and at the side
of A
that faces B
.

noexcept 
Weights for the intruding solution of a neighboring elementcentered subdomain, increasing from 0 to 0.5 towards the side
over the logical distance width
, and further to 1 over the same distance outside the element.

noexcept 
Weights for data on overlap regions intruding into an elementcentered subdomain.
Constructs the weighting field \(W(\xi)\) as described in LinearSolver::Schwarz::element_weight
for the data that overlaps with the central element of an elementcentered subdomain. The weights are constructed in such a way that all weights at a grid point sum to one, i.e. the weight is conserved. The logical_coords
are the elementlogical coordinates of the central element.
This function assumes that corner and edgeneighbors of the central element are not part of the subdomain, which means that no contributions from those neighbors are expected although the weighting field is nonzero in overlap regions with those neighbors. Therefore, to retain conservation we must account for this missing weight by adding it to the central element, to the intruding overlaps from faceneighbors, or split it between the two. We choose to add the weight to the intruding overlaps, since that's where information from the corner and edgeregions propagates through in a DG context.

noexcept 
The number of points that an overlap extends into the volume_extent
In a dimension where an element has volume_extent
points, the overlap extent is the largest number under these constraints:
max_overlap
.volume_extent
.This means the overlap extent is always smaller than the volume_extent
. The reason for this constraint is that we define the width of the overlap as the elementlogical coordinate distance from the face of the element to the first collocation point outside the overlap extent. Therefore, even an overlap region that covers the full element in width does not include the collocation point on the opposite side of the element.
Here's a few notes on the definition of the overlap extent and width:

noexcept 
Total number of grid points in an overlap region that extends overlap_extent
points into the volume_extents
from either side in the overlap_dimension
The overlap region has overlap_extent
points in the overlap_dimension
, and volume_extents
points in the other dimensions. The number of grid points returned by this function is the product of these extents.

noexcept 
Width of an overlap extending overlap_extent
points into the collocation_points
from either side.
The "width" of an overlap is the elementlogical coordinate distance from the element boundary to the first collocation point outside the overlap region in the overlap dimension, i.e. the dimension perpendicular to the element face. See LinearSolver::Schwarz::overlap_extent
for details.
This function assumes the collocation_points
are mirrored around 0.