Analysis of Schwarz methods for a hybridizable discontinuous Galerkin discretization: the many subdomain case
For researchers and practitioners solving large-scale elliptic and parabolic PDEs with HDG discretizations, this work provides the first rigorous convergence estimates for OSM in the many-subdomain case, enabling more efficient parallel solvers.
This paper generalizes optimized Schwarz methods (OSM) for hybridizable discontinuous Galerkin (HDG) discretizations from two subdomains to many subdomains, providing sharp convergence rates with respect to mesh size, polynomial degree, subdomain diameter, and the zeroth-order term of the PDE. These estimates enable precise convergence predictions for OSM applied to parabolic problems with implicit time stepping.
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.