Convergence estimates for multigrid algorithms with SSC smoothers and applications to overlapping domain decomposition
This work advances the theoretical understanding of multigrid methods with SSC smoothers, particularly for problems without regularity assumptions and with local refinement, benefiting researchers in numerical analysis and scientific computing.
The paper provides convergence estimates for multigrid algorithms using successive subspace correction (SSC) smoothers for symmetric elliptic PDEs, deriving explicit error bounds that highlight dependence on smoothing steps and showing level-independent convergence for uniform and local refinement applications with overlapping subdomains.
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.