Preconditioners and Their Analyses for Edge Element Saddle-point Systems Arising from Time-harmonic Maxwell Equations

TL;DR

This paper introduces and analyzes new preconditioners for saddle-point systems from edge element discretizations of time-harmonic Maxwell equations, improving solution efficiency across different wave number regimes.

Contribution

It proposes novel preconditioners and provides a comprehensive analysis of their spectral properties and effectiveness for Maxwell saddle-point systems.

Findings

Preconditioners are effective for small wave numbers using PCG.

Preconditioned MINRES is suitable for large wave numbers.

Numerical experiments confirm the efficiency of the proposed methods.

Abstract

We shall propose and analyze some new preconditioners for the saddle-point systems arising from the edge element discretization of the time-harmonic Maxwell equations in three dimensions. We will first consider the saddle-point systems with vanishing wave number, for which we present an important relation between the solutions of the singular curl-curl system and the non-singular saddle-point system, then demonstrate that the saddle-point system can be efficiently solved by the Hiptmair-Xu solver. For the saddle-point systems with non-vanishing wave numbers, we will show that the PCG with a new preconditioner can apply for the non-singular system when wave numbers are small, while the methods like preconditioned MINRES may apply for some existing and new preconditioners when wave numbers are large. The spectral behaviors of the resulting preconditioned systems for the existing and new preconditioners are analyzed and compared, and numerical experiments are presented to demonstrate and compare the efficiencies of these preconditioners.