Analysis of Schwarz methods for a hybridizable discontinuous Galerkin discretization: the many subdomain case

TL;DR

This paper extends the analysis of optimized Schwarz methods (OSM) for hybridizable discontinuous Galerkin (HDG) discretizations from two subdomains to many, providing sharp convergence estimates for large-scale PDE problems.

Contribution

It generalizes the convergence analysis of OSM for HDG discretizations to multiple subdomains, including sharp estimates related to mesh size, polynomial degree, and PDE parameters.

Findings

Sharp convergence rates with respect to mesh size and polynomial degree

Effective application of OSM to parabolic problems with implicit time stepping

Numerical experiments validating theoretical convergence estimates

Abstract

Schwarz methods are attractive parallel solution techniques for solving large-scale linear systems obtained from discretizations of partial differential equations (PDEs). Due to the iterative nature of Schwarz methods, convergence rates are an important criterion to quantify their performance. Optimized Schwarz methods (OSM) form a class of Schwarz methods that are designed to achieve faster convergence rates by employing optimized transmission conditions between subdomains. It has been shown recently that for a two-subdomain case, OSM is a natural solver for hybridizable discontinuous Galerkin (HDG) discretizations of elliptic PDEs. In this paper, we generalize the preceding result to the many-subdomain case and obtain sharp convergence rates with respect to the mesh size and polynomial degree, the subdomain diameter, and the zeroth-order term of the underlying PDE, which allows us for the first time to give precise convergence estimates for OSM used to solve parabolic problems by implicit time stepping. We illustrate our theoretical results with numerical experiments.