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

TL;DR

This work develops and numerically investigates two-level overlapping Schwarz preconditioners using GDSW coarse spaces for Virtual Element Method (VEM) discretizations of the Poisson problem on polygonal meshes. The authors implement GDSW, GDSW*, and RGDSW variants in a PETSc-based framework and couple them with the Vem++ library, showcasing the first parallel application of these coarse-space preconditioners to VEM. Numerical experiments in 2D and 3D with polynomial degrees $k=1,2$ on random Voronoi meshes demonstrate strong scalability up to 1000 cores and reveal substantial coarse-space reductions for RGDSW and GDSW* without sacrificing convergence. The results highlight the robustness and practicality of GDSW-type coarse spaces for large-scale VEM simulations and point to future work on higher-order extensions, convergence theory, and broader model problems such as linear elasticity.

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.

Figures (1)

• Figure 1: Domain decomposition of randomly seeded Voronoi meshes: (left) 2D decomposition into 4$\times$4 subdomains of a Voronoi mesh with 10 000 elements; (right) 3D decomposition into 4$\times$4$\times$4 subdomains of a Voronoi mesh with 16 000 elements.