Spectral Element Methods for Eigenvalue Problems Based on Domain Decomposition

Abstract

The spectral element method performs very well in solving PDE eigenvalue problems for some cases. In this paper, we propose and analyze parallel spectral element methods to solve elliptic PDE eigenvalue problems (although we use the Laplacian operator as a model) based on domain decomposition methods. The method demonstrates great advantage in both high accuracy and high efficiency by combining the spectral element method and parallel computing. In addition, the method could be performed in parallel with good scalability. We present a systematic construction and analysis for both the non-shifted scheme and the shifted scheme in 1D, 2D, and 3D. Furthermore, we obtain an optimal convergence rate of the eigenpair in the sense that it is independent of $h$ (the size of the spectral elements), $p_i$ (the degree of the polynomial in each interior spectral element $T_i$ of subdomains), and $N$ (the number of subdomains) by choosing a suitable overlapping size $\delta$. Numerical tests are provided to show the efficiency of the method.