Fast Maxwell Solvers Based on Exact Discrete Eigen-Decompositions I. Two-Dimensional Case

TL;DR

The paper develops fast solvers for two-dimensional Maxwell curl-curl equations on rectangular domains by discretizing with the lowest-order Nédélec edge elements and constructing exact discrete eigen-decompositions. This yields a discrete Helmholtz-Hodge decomposition that splits the approximation space into divergence-free and curl-free components, allowing solutions to be expressed in terms of divergence-free eigenfunctions and solved efficiently with discrete sine/cosine transforms to achieve $O(n^2 \log n)$ complexity. The method preserves the divergence-free constraint or Gauss's law at the discrete level and serves as an effective preconditioner for variable-coefficient problems, outperforming traditional LU-based direct solvers in time and storage. Extensions to natural boundary conditions, general Maxwell formulations, and potential three-dimensional solvers are discussed, with numerical experiments confirming accuracy, efficiency, and robustness across multiple scenarios.

Abstract

In this paper, we propose fast solvers for Maxwell's equations in rectangular domains. We first discretize the simplified Maxwell's eigenvalue problems by employing the lowest-order rectangular Nédélec elements and derive the discrete eigen-solutions explicitly, providing a Hodge-Helmholtz decomposition framework at the discrete level. Based on exact eigen-decompositions, we further design fast solvers for various Maxwell's source problems, guaranteeing either the divergence-free constraint or the Gauss's law at the discrete level. With the help of fast sine/cosine transforms, the computational time grows asymptotically as $\mathcal{O}(n^2\log n)$ with $n$ being the number of grids in each direction. Our fast Maxwell solvers outperform other existing Maxwell solvers in the literature and fully rival fast scalar Poisson/Helmholtz solvers based on trigonometric transforms in either efficiency, robustness, or storage complexity. It is also utilized to perform an efficient pre-conditioning for solving Maxwell's source problems with variable coefficients. Finally, numerical experiments are carried out to illustrate the effectiveness and efficiency of the proposed fast solver.

Figures (7)

• Figure 7.1: The number of non-zeros $n_{\rm DM}$ and $n_{\rm LU}$ versus $n$ in Example \ref{['exam1']}.
• Figure 7.2: Total computational time versus $n$ in Example \ref{['exam1']}.
• Figure 7.3: Total computational time versus $n$ in Example \ref{['exam2']}.
• Figure 7.4: Total computational time versus $n$ in Example \ref{['exam3']}.
• Figure 7.5: Iterations versus $n$ for CG and PCG in Example \ref{['exam4']} with $\varepsilon=10^{-14}$.

Theorems & Definitions (15)

• Lemma 2.1
• Theorem 3.1
• Remark 4.1
• Remark 4.2
• Remark 4.3
• Lemma 5.1
• Theorem 5.1
• Remark 5.1
• Remark 5.2
• Remark 6.1