Parallel solvers for virtual element discretizations of elliptic equations in mixed form

TL;DR

This paper evaluates the performance of mixed virtual element methods for 3D elliptic equations on polyhedral meshes and investigates parallel solution strategies for the resulting linear systems, including new preconditioners.

Contribution

It introduces the first numerical testing of 3D mixed virtual elements and develops two novel parallel block preconditioners for the associated linear systems.

Findings

VEM discretization achieves expected convergence rates.

Parallel solvers demonstrate scalable performance on Linux clusters.

Preconditioners improve iterative solver efficiency.

Abstract

The aim of this paper is twofold. On the one hand, we test numerically the performance of mixed virtual elements in three dimensions for the first time in the literature to solve the mixed formulation of three-dimensional elliptic equations on polyhedral meshes. On the other hand, we focus on the parallel solution of the linear system arising from such discretization, considering both direct and iterative parallel solvers. In the latter case, we develop two block preconditioners, one based on the approximate Schur complement and one on a regularization technique. Both these topics are numerically validated by several parallel tests performed on a Linux cluster. More specifically, we show that the proposed VEM discretization recovers the expected theoretical convergence properties and we analize the performance of the direct and iterative parallel solvers taken into account.