Robust Solvers for Maxwell's Equations with Dissipative Boundary Conditions

TL;DR

This paper develops robust, structure-preserving linear solvers for Maxwell's equations with dissipative boundary conditions, using finite-element discretization and block preconditioners, validated by numerical tests.

Contribution

Introduces two novel block preconditioners for Maxwell's equations that are proven robust and optimal, leveraging structure-preserving discretization.

Findings

Preconditioners are robust across various problem parameters.

Numerical tests confirm optimality and efficiency.

Structure-preserving discretization leads to sparse Schur complements.

Abstract

In this paper, we design robust and efficient linear solvers for the numerical approximation of solutions to Maxwell's equations with dissipative boundary conditions. We consider a structure-preserving finite-element approximation with standard Nedelec--Raviart--Thomas elements in space and a Crank--Nicolson scheme in time to approximate the electric and magnetic fields. We focus on two types of block preconditioners. The first type is based on the well-posedness results of the discrete problem. The second uses an exact block factorization of the linear system, for which the structure-preserving discretization yields sparse Schur complements. We prove robustness and optimality of these block preconditioners, and provide supporting numerical tests.