Convergence analysis for GMsFEM approximation of elliptic eigenvalue problems

In this paper, we analyze the approximation of elliptic eigenvalue problems using generalized multiscale finite element method (GMsFEM) and get error estimates for eigenfunctions and eigenvalues. For the case of simple eigenvalues, the approximation errors for eigenfunctions are considered in both energy error and L2 error. The derived error estimates clearly give the relation between the errors and the coarse mesh size, local multiscale enrichment and the corresponding eigenvalues. The convergence analysis shows that the approximation of eigenvalue problems using GMsFEM does not depend on the contrastness of the coefficient when the diffusion coefficient is highly heterogeneous. A few numerical examples are presented to illustrate the performance of the GMsFEM approximation and the theoretic analysis for the eigenvalue problems.