Convergence estimates for multigrid algorithms with SSC smoothers and applications to overlapping domain decomposition

TL;DR

This paper provides new convergence estimates for multigrid algorithms with SSC smoothers applied to elliptic PDEs, including cases with irregular solutions and complex grid refinements, enhancing theoretical understanding and practical applicability.

Contribution

It introduces a general convergence analysis for multigrid with SSC smoothers without regularity assumptions and derives bounds for overlapping domain decompositions, including local refinement with hanging nodes.

Findings

Explicit error bounds depending on smoothing steps

Conditions for multigrid convergence independent of levels

Applicability to complex grid refinement scenarios

Abstract

In this paper we study convergence estimates for a multigrid algorithm with smoothers of successive subspace correction (SSC) type, applied to symmetric elliptic PDEs. First, we revisit a general convergence analysis on a class of multigrid algorithms in a fairly general setting, where no regularity assumptions are made on the solution. In this framework, we are able to explicitly highlight the dependence of the multigrid error bound on the number of smoothing steps. For the case of no regularity assumptions, this represents a new addition to the existing theory. Then, we analyze successive subspace correction smoothing schemes for a set of uniform and local refinement applications with either nested or non-nested overlapping subdomains. For these applications, we explicitly derive bounds for the multigrid error, and identify sufficient conditions for these bounds to be independent of the number of multigrid levels. For the local refinement applications, finite element grids with arbitrary hanging nodes configurations are considered. The analysis of these smoothing schemes is cast within the far-reaching multiplicative Schwarz framework.