An Augmented Two-Scale Finite Element Method for Eigenvalue Problems

In this paper, an augmented two-scale finite element method is proposed for a class of linear and nonlinear eigenvalue problems on tensor-product domains. Through a correction step, the augmented two-scale finite element solution is obtained by solving an eigenvalue problem on a low-dimensional augmented subspace. Theoretical analysis and numerical experiments show that the augmented two-scale finite element solution achieves the same order of accuracy as the standard finite element solution on a fine grid, but the computational cost required by the former solution is much lower than that demanded by the latter. The augmented two-scale finite element method also improves the approximation accuracy of eigenfunctions in the L2(Omega)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L<^>2(\varOmega )$$\end{document} norm compared with the two-scale finite element method.