Parallel block preconditioners for virtual element discretizations of the time-dependent Maxwell equations

TL;DR

This paper develops and validates parallel block preconditioners for virtual element discretizations of 3D Maxwell equations, improving the efficiency of solving large, ill-conditioned linear systems in time-dependent simulations.

Contribution

It introduces a novel parallel preconditioner leveraging Schur complement factorization tailored for VEM discretizations of Maxwell equations.

Findings

Preconditioner is robust across mesh refinements

Effective for various polyhedral element shapes

Maintains performance with different time step sizes

Abstract

The focus of this study is the construction and numerical validation of parallel block preconditioners for low order virtual element discretizations of the three-dimensional Maxwell equations. The virtual element method (VEM) is a recent technology for the numerical approximation of partial differential equations (PDEs), that generalizes finite elements to polytopal computational grids. So far, VEM has been extended to several problems described by PDEs, and recently also to the time-dependent Maxwell equations. When the time discretization of PDEs is performed implicitly, at each time-step a large-scale and ill-conditioned linear system must be solved, that, in case of Maxwell equations, is particularly challenging, because of the presence of both H(div) and H(curl) discretization spaces. We propose here a parallel preconditioner, that exploits the Schur complement block factorization of the linear system matrix and consists of a Jacobi preconditioner for the H(div) block and an auxiliary space preconditioner for the H(curl) block. Several parallel numerical tests have been perfomed to study the robustness of the solver with respect to mesh refinement, shape of polyhedral elements, time step size and the VEM stabilization parameter.