(untitled) (CSIAM Transactions on Applied Mathematics, 2025)
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 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 O(N^6) and O(N^9) to O(N^log_2 7) and 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.