Coarse spaces for virtual element methods on irregular 3D subdomain decompositions

TL;DR

This paper introduces a two-level overlapping Schwarz preconditioner for 3D virtual element methods on irregular polyhedral meshes, demonstrating robustness and simplicity compared to existing methods.

Contribution

It extends overlapping Schwarz preconditioners to handle general polyhedral meshes and irregular subdomains in 3D virtual element discretizations.

Findings

Robust performance with respect to subdomain number and mesh parameters

Condition-number bounds comparable to classical finite element methods

Simpler and competitive alternative to FETI-DP and BDDC methods

Abstract

We present a two-level overlapping Schwarz preconditioner for three-dimensional problems discretized with the Virtual Element Method. Our approach handles general polyhedral meshes and irregular subdomains, extending the applicability of previous methods. Numerical experiments show robust performance with respect to the number of subdomains and mesh parameters, with condition-number bound comparable to classical finite element results. While alternative methods such as FETI-DP and BDDC are available, the simplicity and competitiveness of overlapping additive Schwarz methods underscore the practical significance of our contribution.