Highly efficient Gauss's lawpreserving spectral algorithms for Maxwell's doublecurl source and eigenvalue problems based on eigendecomposition
Abstract
In this paper, we present Gauss's lawpreserving spectral methods and their efficient solution algorithms for curlcurl 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 saddlepoint nature of the mixed scheme, we present highly efficient solution algorithms based on reordering and decoupling of the resultant linear algebraic system and numerical eigendecomposition of one dimensional mass matrix. The proposed solution algorithms are direct methods requiring only several matrixmatrix or matrixtensor 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 prefactors 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 HelmholtzHodge decomposition, thus totally eliminate the spurious eigenmodes of nonphysical 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.
 Publication:

arXiv eprints
 Pub Date:
 February 2024
 DOI:
 10.48550/arXiv.2402.19125
 arXiv:
 arXiv:2402.19125
 Bibcode:
 2024arXiv240219125L
 Keywords:

 Mathematics  Numerical Analysis