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

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 $\mathbf{H}{\rm (curl)}$-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 $\mathcal{O}(N^6)$ and $\mathcal{O}(N^9)$ to $\mathcal{O}(N^{log_2 7})$ and $\mathcal{O}(N^{1+log_2 7})$ 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.