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.

中文翻译：

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基于特征分解的麦克斯韦双旋源及特征值问题的高效高斯保律谱算法

在本文中，我们提出了高斯保律谱方法及其有效的求解算法，用于解决麦克斯韦方程组产生的二维和三维旋度-旋度源和特征值问题。首先使用勒让德多项式的紧凑组合提出了二维和三维的任意阶 $H(curl)$ 一致谱基函数。然后采用涉及拉格朗日乘子的混合公式来保留弱意义上的高斯定律。为了克服混合方案的鞍点性质引起的计算效率瓶颈，我们提出了基于所得线性代数系统的重排序和解耦以及一维质量矩阵的数值特征分解的高效求解算法。所提出的求解算法是直接方法，仅需要 $N$×$N$ 矩阵的几个矩阵-矩阵或矩阵-张量乘积，其中 $N$ 是每个方向上的最高多项式阶数。与其他直接方法相比，计算复杂度从 $O(N^6)$ 和 $O(N^9)$ 降低到 $O(N^3)$ 和 $O(N^4)$，且计算量较小分别适用于 2D 和 3D 情况的常数前因子，并且当使用 Strassen 矩阵乘法算法增强时，可以进一步加速到 $O(N^{2.807})$ 和 $O(N^{3.807})$。而且，这些算法严格遵循Helmholtz-Hodge分解，从而完全消除了非物理零特征值的寄生特征模。讨论了所提出的方法和算法对具有可变系数和不均匀边界条件的复杂几何问题的扩展，以处理更一般的情况。给出了解决麦克斯韦源和特征值问题的大量数值例子，以证明所提出方法的准确性和效率。