Towards a parallel Schwarz solver framework for virtual elements using GDSW coarse spaces

TL;DR

This paper develops and tests a parallel Schwarz solver framework using GDSW coarse spaces for the Virtual Element Method, demonstrating scalability on large parallel systems for 2D and 3D problems.

Contribution

It introduces the first parallel implementation of GDSW preconditioners for VEM and evaluates their performance and scalability.

Findings

Scalability demonstrated up to 1,000 cores

Effective parallel implementation with PETSc and Vem++

Suitable for high-degree VEM discretizations

Abstract

The Virtual Element Method (VEM) is used to perform the discretization of the Poisson problem on polygonal and polyhedral meshes. This results in a symmetric positive definite linear system, which is solved iteratively using overlapping Schwarz domain decomposition preconditioners, where to ensure robustness and parallel scalability a second level has to be employed. The construction and numerical study of two-level overlapping Schwarz preconditioners with variants of the GDSW (Generalized Dryja-Smith-Widlund) coarse space are presented here. Our PETSc-based parallel implementation of GDSW and variants, combined with the Vem++ library, represent the first parallel application of these GDSW preconditioners to VEM. Numerical experiments in 2D and 3D demonstrate scalability of our preconditioners up to 1 000 parallel cores for VEM discretizations of degrees k=1,2.