Highly Efficient Gauss’s Law-Preserving Spectral Algorithms for Maxwell’s Double-Curl Source and Eigenvalue Problems Based on Eigen-Decomposition (CSIAM Transactions on Applied Mathematics, 2025)
In this paper, we present the Gauss’s law-preserving spectral methods andtheir efficient solution algorithms for curl-curl source and eigenvalue problems arisingfrom Maxwell’s equations. Arbitrary order $H{\rm (curl)}$-conforming spectral basis functions in two and three dimensions are firstly proposed using compact combination ofLegendre polynomials. A mixed formulation involving a Lagrange multiplier is thenadopted to preserve the Gauss’s law in the weak sense. To overcome the bottleneck ofcomputational efficiency caused by the saddle-point nature of the mixed scheme, wepresent highly efficient algorithms based on reordering and decoupling of the linearsystem and numerical eigen-decomposition of 1D mass matrix. The proposed solutionalgorithms are direct methods requiring only several matrix-matrix or matrix-tensorproducts 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 constantpre-factors for 2D and 3D cases, respectively. Moreover, these algorithms strictly obeythe Helmholtz-Hodge decomposition, thus totally eliminate the spurious eigen-modesof non-physical zero eigenvalues for convex domains. Ample numerical examples forsolving Maxwell’s source and eigenvalue problems are presented to demonstrate theaccuracy and efficiency of the proposed methods.
Keywords: Spectral method; time-harmonic Maxwell’s equation; structure-preserving method; Gauss’s law preservation; fast solver.