Highly Efficient Gauss's Law-Preserving Spectral Algorithms for Maxwell's Double-Curl Source and Eigenvalue Problems Based on Eigen-Decomposition

Abstract

In this paper, we present the Gauss's law-preserving spectral methods and their efficient solution algorithms for curl-curl source and eigenvalue problems arising from Maxwell's equations. Arbitrary order $\mathit{H}$ (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 algorithms based on reordering and decoupling of the linear system and numerical eigen-decomposition of 1D 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 $\mathcal{O}\left({N}^{6}\right)$ and $\mathcal{O}\left({N}^{9}\right)$ to $\mathcal{O}\left({N}^{{\mathrm{l}\mathrm{o}\mathrm{g}}_{2}7}\right)$ and $\mathcal{O}\left({N}^{1+{\mathrm{l}\mathrm{o}\mathrm{g}}_{2}7}\right)$ with small and constant pre-factors for 2D and 3D cases, respectively. Moreover, these algorithms strictly obey the Helmholtz-Hodge decomposition, thus totally eliminate the spurious eigen-modes of non-physical zero eigenvalues for convex domains. Ample numerical examples for solving Maxwell's source and eigenvalue problems are presented to demonstrate the accuracy and efficiency of the proposed methods.

Keywords

Spectral method / time-harmonic Maxwell's equation / structure-preserving method / Gauss's law preservation / fast solver