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A two-level domain-decomposition preconditioner for the time-harmonic Maxwell's equations

Marcella Bonazzoli, Victorita Dolean, Ivan Graham, Euan Spence, Pierre-Henri Tournier

TL;DR

This work extends two-level domain-decomposition preconditioners, originally developed for the high-frequency Helmholtz equation, to the time-harmonic Maxwell system with absorption. By formulating a variational problem in $H_0(\mathrm{curl};\Omega)$ and discretizing with Nédélec elements, it constructs local and coarse spaces and defines AS, RAS, and impedance-based HRAS variants, along with their theoretical and numerical analyses. A Maxwell-specific GMRES convergence result shows parameter-robust behavior when $\kappa \sim k^2$ under generous coarse-subdomain overlap, and numerical experiments demonstrate robust convergence and favorable scaling across PEC and impedance boundaries, highlighting ImpHRAS as a particularly effective choice in many settings. The findings indicate that two-level, absorption-informed domain-decomposition preconditioners enable efficient, scalable solvers for mid-to-high-frequency Maxwell problems.

Abstract

The construction of fast iterative solvers for the indefinite time-harmonic Maxwell's system at mid- to high-frequency is a problem of great current interest. Some of the difficulties that arise are similar to those encountered in the case of the mid- to high-frequency Helmholtz equation. Here we investigate how two-level domain-decomposition preconditioners recently proposed for the Helmholtz equation work in the Maxwell case, both from the theoretical and numerical points of view.

A two-level domain-decomposition preconditioner for the time-harmonic Maxwell's equations

TL;DR

This work extends two-level domain-decomposition preconditioners, originally developed for the high-frequency Helmholtz equation, to the time-harmonic Maxwell system with absorption. By formulating a variational problem in and discretizing with Nédélec elements, it constructs local and coarse spaces and defines AS, RAS, and impedance-based HRAS variants, along with their theoretical and numerical analyses. A Maxwell-specific GMRES convergence result shows parameter-robust behavior when under generous coarse-subdomain overlap, and numerical experiments demonstrate robust convergence and favorable scaling across PEC and impedance boundaries, highlighting ImpHRAS as a particularly effective choice in many settings. The findings indicate that two-level, absorption-informed domain-decomposition preconditioners enable efficient, scalable solvers for mid-to-high-frequency Maxwell problems.

Abstract

The construction of fast iterative solvers for the indefinite time-harmonic Maxwell's system at mid- to high-frequency is a problem of great current interest. Some of the difficulties that arise are similar to those encountered in the case of the mid- to high-frequency Helmholtz equation. Here we investigate how two-level domain-decomposition preconditioners recently proposed for the Helmholtz equation work in the Maxwell case, both from the theoretical and numerical points of view.

Paper Structure

This paper contains 5 sections, 1 theorem, 16 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Variational formulation and discretisation
  3. Domain decomposition
  4. Theoretical results
  5. Numerical results

Key Result

theorem 1

Assume that $\Omega$ is a convex polyhedron. Let $C_k$ be the matrix representing the $(\cdot, \cdot)_{\text{curl},k}$ inner product on the finite element space ${\mathcal{V}}^h$ in the sense that if $v_h, w_h \in {\mathcal{V}}^h$ with coefficient vectors $\mathbf{V}, \mathbf{W}$ then Consider the weighted GMRES method where the residual is minimised in the norm induced by $C_k$. Let $\mathbf{r}^

Theorems & Definitions (1)

  • theorem 1: GMRES convergence for left preconditioning with $\kappa\sim k^2$