Hierarchical LU preconditioning for the time-harmonic Maxwell equations

TL;DR

This paper explores hierarchical LU preconditioning techniques for efficiently solving the indefinite linear systems arising from the finite element discretization of time-harmonic Maxwell equations, aiming to improve convergence of iterative solvers.

Contribution

It introduces the use of $ ext{H-}LU$ preconditioning combined with $ ext{H-} ext{matrix}$ approximations to enhance the solution process for Maxwell equations discretized by Nédélec's FEM.

Findings

Hierarchical LU preconditioning reduces iteration counts.

Inverse $ ext{H-}$matrix approximations improve solver efficiency.

Preconditioning enhances convergence for indefinite Maxwell systems.

Abstract

The time-harmonic Maxwell equations are used to study the effect of electric and magnetic fields on each other. Although the linear systems resulting from solving this system using FEMs are sparse, direct solvers cannot reach the linear complexity. In fact, due to the indefinite system matrix, iterative solvers suffer from slow convergence. In this work, we study the effect of using the inverse of $\CH$-matrix approximations of the Galerkin matrices arising from N\'ed\'elec's edge FEM discretization to solve the linear system directly. We also investigate the impact of applying an $\mathcal{H}-LU$ factorization as a preconditioner and we study the number of iterations to solve the linear system using iterative solvers.