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.

翻译：本文针对Maxwell方程中二维和三维旋度-旋度源问题及特征值问题，提出了保持高斯定律的谱方法及其高效求解算法。首先利用Legendre多项式的紧致组合构造了任意阶二维和三维$H(curl)$共形谱基函数。随后采用含拉格朗日乘子的混合形式，在弱意义上保持高斯定律。为克服混合格式鞍点性质导致的计算效率瓶颈，本文基于线性代数系统的重排序与解耦策略，以及一维质量矩阵的数值特征分解，提出了高效求解算法。所提算法为直接方法，仅需若干$N\times N$矩阵与矩阵或矩阵与张量的乘积运算（$N$为各方向最高多项式阶数）。相较于其他直接法，二维和三维情况下的计算复杂度分别从$O(N^6)$和$O(N^9)$降至$O(N^3)$和$O(N^4)$，且预因子为小常数；若采用Strassen矩阵乘法加速，可进一步降低至$O(N^{2.807})$和$O(N^{3.807})$。此外，该算法严格满足Helmholtz-Hodge分解，从而完全消除非物理零特征值的伪特征模态。本文进一步讨论了所提方法在复杂几何区域、变系数及非齐次边界条件问题中的推广，以处理更一般的情形。大量Maxwell源问题与特征值问题的数值算例验证了所提方法的精度与效率。