A V-cycle geometric multgrid solver for linear equations \(Ax = b\). More...

#include <Multigrid.hpp>

Public Types
using	fields_tag = FieldsTag

using	options_group = OptionsGroup

using	source_tag = SourceTag

using	operand_tag = FieldsTag

using	smooth_source_tag = source_tag

using	smooth_fields_tag = fields_tag

using	component_list = tmpl::list<>

using	observed_reduction_data_tags = observers::make_reduction_data_tags< tmpl::list< async_solvers::reduction_data > >

using	initialize_element = tmpl::list< async_solvers::InitializeElement< FieldsTag, OptionsGroup, SourceTag >, detail::InitializeElement< Dim, FieldsTag, OptionsGroup, SourceTag > >

using	register_element = tmpl::list< async_solvers::RegisterElement< FieldsTag, OptionsGroup, SourceTag, Tags::IsFinestGrid >, observers::Actions::RegisterWithObservers< detail::RegisterWithVolumeObserver< OptionsGroup > > >

template<typename PreSmootherActions , typename PostSmootherActions , typename Label = OptionsGroup>
using	solve = tmpl::list< async_solvers::PrepareSolve< FieldsTag, OptionsGroup, SourceTag, Label, Tags::IsFinestGrid, false >, detail::ReceiveResidualFromFinerGrid< Dim, FieldsTag, OptionsGroup, SourceTag >, detail::PreparePreSmoothing< FieldsTag, OptionsGroup, SourceTag >, PreSmootherActions, detail::SkipPostsmoothingAtBottom< FieldsTag, OptionsGroup, SourceTag >, detail::SendResidualToCoarserGrid< FieldsTag, OptionsGroup, ResidualIsMassiveTag, SourceTag >, detail::ReceiveCorrectionFromCoarserGrid< Dim, FieldsTag, OptionsGroup, SourceTag >, PostSmootherActions, detail::SendCorrectionToFinerGrid< FieldsTag, OptionsGroup, SourceTag >, detail::ObserveVolumeData< FieldsTag, OptionsGroup, SourceTag >, async_solvers::CompleteStep< FieldsTag, OptionsGroup, SourceTag, Label, Tags::IsFinestGrid, false > >

Detailed Description

template<size_t Dim, typename FieldsTag, typename OptionsGroup, typename ResidualIsMassiveTag, typename SourceTag = db::add_tag_prefix<::Tags::FixedSource, FieldsTag>>
struct LinearSolver::multigrid::Multigrid< Dim, FieldsTag, OptionsGroup, ResidualIsMassiveTag, SourceTag >

A V-cycle geometric multgrid solver for linear equations \(Ax = b\).

This linear solver iteratively corrects an initial guess \(x_0\) by restricting the residual \(b - Ax\) to a series of coarser grids, solving for a correction on the coarser grids, and then prolongating (interpolating) the correction back to the finer grids. The solves on grids with different scales can very effectively solve large-scale modes in the solution, which Krylov-type linear solvers such as GMRES or Conjugate Gradients typically struggle with. Therefore, a multigrid solver can be an effective preconditioner for Krylov-type linear solvers (see LinearSolver::gmres::Gmres and LinearSolver::cg::ConjugateGradient). See [19] for an introduction to multigrid methods.

Grid hierarchy: This geometric multigrid solver relies on a strategy to coarsen the computational grid in a way that removes small-scale modes. We currently h-coarsen the domain, meaning that we create multigrid levels by successively combining two elements into one along every dimension of the grid. We only p-coarsen the grid in the sense that we choose the smaller of the two polynomial degrees when combining elements. This strategy follows [110]. See LinearSolver::multigrid::ElementsAllocator and LinearSolver::multigrid::coarsen for the code that creates the multigrid hierarchy.

Inter-mesh operators: The algorithm relies on operations that project data between grids. Residuals are projected from finer to coarser grids ("restriction") and solutions are projected from coarser to finer grids ("prolongation"). We use the standard \(L_2\)-projections implemented in Spectral::projection_matrix_child_to_parent and Spectral::projection_matrix_parent_to_child, where "child" means the finer grid and "parent" means the coarser grid. Note that the residuals \(b - Ax\) may or may not already include a mass matrix and Jacobian factors, depending on the implementation of the linear operator \(A\). To account for this, provide a tag as the ResidualIsMassiveTag that holds a bool. See section 3 in [44] for details on the inter-mesh operators.

Smoother: On every level of the grid hierarchy the multigrid solver relies on a "smoother" that performs an approximate linear solve on that level. The smoother can be any linear solver that solves the smooth_fields_tag for the source in the smooth_source_tag. Useful smoothers are asynchronous linear solvers that parallelize well, such as LinearSolver::Schwarz::Schwarz. The multigrid algorithm doesn't assume anything about the smoother, but requires only that the PreSmootherActions and the PostSmootherActions passed to solve leave the smooth_fields_tag in a state that represents an approximate solution to the smooth_source_tag, and that db::add_tag_prefix<LinearSolver::Tags::OperatorAppliedTo, smooth_fields_tag> is left up-to-date as well. Here's an example of setting up a smoother for the multigrid solver:

  using multigrid = LinearSolver::multigrid::Multigrid<
      volume_dim, helpers_mg::fields_tag, MultigridSolver,
      helpers_mg::OperatorIsMassive, helpers_mg::sources_tag>;
 
  using smoother = LinearSolver::Richardson::Richardson<
      typename multigrid::smooth_fields_tag, RichardsonSmoother,
      typename multigrid::smooth_source_tag,
      LinearSolver::multigrid::Tags::MultigridLevel>;

The smoother can be used to construct an action list like this:

  template <typename Label>
  using smooth_actions = tmpl::list<
      helpers_mg::ComputeOperatorAction<typename smoother::fields_tag>,
      typename smoother::template solve<
          helpers_mg::ComputeOperatorAction<typename smoother::operand_tag>,
          Label>>;
 
  using solve_actions =
      tmpl::list<typename multigrid::template solve<
                     smooth_actions<LinearSolver::multigrid::VcycleDownLabel>,
                     smooth_actions<LinearSolver::multigrid::VcycleUpLabel>>,
                 Parallel::Actions::TerminatePhase>;

Algorithm overview: Every iteration of the multigrid algorithm performs a V-cycle over the grid hierarchy. One V-cycle consists of first "going down" the grid hierarchy, from the finest to successively coarser grids, smoothing on every level ("pre-smoothing"), and then "going up" the grid hierarchy again, smoothing on every level again ("post-smoothing"). When going down, the algorithm projects the remaining residual of the smoother to the next-coarser grid, setting it as the source for the smoother on the coarser grid. When going up again, the algorithm projects the solution of the smoother to the next-finer grid, adding it to the solution on the finer grid as a correction. The bottom-most coarsest grid (the "tip" of the V-cycle) skips the post-smoothing, so the result of the pre-smoother is immediately projected up to the finer grid. On the top-most finest grid (the "original" grid that represents the overall solution) the algorithm applies the smoothing and the corrections from the coarser grids directly to the solution fields.

Public Types

Detailed Description