Highly efficient Gauss's law-preserving spectral algorithms for Maxwell's double-curl source and eigenvalue problems based on eigen-decomposition

TL;DR

The paper addresses the numerical solution of Maxwell's time-harmonic double-curl problems with Gauss's law constraints by developing Gauss's law-preserving spectral methods that use arbitrary-order $H({\rm curl})$-conforming bases built from Legendre polynomials and a Kikuchi mixed formulation. A novel matrix-free, semi-analytic solver decouples the resulting saddle-point systems via eigen-decomposition of the 1D mass matrix, achieving significant reductions in computational complexity in both 2D ($\mathcal{O}(N^3)$ or $\mathcal{O}(N^{\log_2 7})$ with Strassen) and 3D ($\mathcal{O}(N^4)$ or $\mathcal{O}(N^{\log_2 14})$ with Strassen). The method preserves the Helmholtz–Hodge decomposition, eliminating spurious zero eigenmodes, and provides exact divergence constraints at the discrete level for nonzero eigenvalues. The authors demonstrate exponential convergence and substantial scalability across 2D and 3D problems, including extensions to non-homogeneous boundaries, complex geometries, and variable coefficients, with numerical results comparing favorably against traditional Yee schemes and showcasing robust eigenvalue behavior (e.g., a predictable fraction of trusted eigenvalues).

Abstract

In this paper, we present Gauss's law-preserving spectral methods and their efficient solution algorithms for curl-curl source and eigenvalue problems in two and three dimensions arising from Maxwell's equations. Arbitrary order $H(curl)$-conforming spectral basis functions in two and three dimensions are firstly proposed using compact combination of Legendre polynomials. A mixed formulation involving a Lagrange multiplier is then adopted to preserve the Gauss's law in the weak sense. To overcome the bottleneck of computational efficiency caused by the saddle-point nature of the mixed scheme, we present highly efficient solution algorithms based on reordering and decoupling of the resultant linear algebraic system and numerical eigen-decomposition of one dimensional mass matrix. The proposed solution algorithms are direct methods requiring only several matrix-matrix or matrix-tensor products of $N$-by-$N$ matrices, where $N$ is the highest polynomial order in each direction. Compared with other direct methods, the computational complexities are reduced from $O(N^6)$ and $O(N^9)$ to $O(N^3)$ and $O(N^4)$ with small and constant pre-factors for 2D and 3D cases, respectively, and can further be accelerated to $O(N^{2.807})$ and $O(N^{3.807})$, when boosted with the Strassen's matrix multiplication algorithm. Moreover, these algorithms strictly obey the Helmholtz-Hodge decomposition, thus totally eliminate the spurious eigen-modes of non-physical zero eigenvalues. Extensions of the proposed methods and algorithms to problems in complex geometries with variable coefficients and inhomogeneous boundary conditions are discussed to deal with more general situations. Ample numerical examples for solving Maxwell's source and eigenvalue problems are presented to demonstrate the accuracy and efficiency of the proposed methods.

Figures (9)

• Figure 1: Example \ref{['Exmp5-1']}: convergence and computational cost tests in 2D. (a) $L^2$- and $H(\rm curl)$-errors of $\boldsymbol{u}$ versus polynomial order $N$ for $\kappa = \pm{100}$; (b) comparison on the computational time versus DoFs for $\kappa=100$ of the direct, Yee and the proposed method.
• Figure 2: Example \ref{['Exmp5-2']}: the profiles of the component $\mathbf{U}$ of the numerical solution versus different $\kappa$. These results are obtained with a fixed polynomial order $N=512$.
• Figure 3: Example \ref{['Exmp5-3']}: (a) $L^2$-errors of $\mathbf{U}$ and $\mathbf{V}$ versus $N$; (b-d) the profiles of $\alpha(\boldsymbol{x})$ and two components $\mathbf{U}$ and $\mathbf{V}$ of the numerical solution with $\kappa=-400$ for $N=200$.
• Figure 4: Example \ref{['Exmp5-4']}: convergence and comparison tests of curl-curl problem in 3D. (a) $L^2$- and $H(\rm curl)$-errors of $\boldsymbol{u}$ versus polynomial order $N$ for $\kappa = \pm{100}$; (b) comparison on the computational time versus DoFs for $\kappa=100$ of the direct and our proposed method.
• Figure 5: Example \ref{['Exmp5-5']}: $L^2$- and $H(\rm curl)$-errors of $\boldsymbol{u}$ versus polynomial order $N$ for $\kappa=-500$.

Theorems & Definitions (2)

• proof
• proof