A Multilevel Correction Scheme Based On The Shifted Inverse Iteration For The Steklov Eigenvalue Problem

Steklov eigenvalue problems have important physical background and wide applications especially in ?uid, thus ?nite element methods for Steklov eigenvalue problems have attracted the attention of more and more scholars. A multilevel method based on the correction step was proposed by Lin and Xie. The shifted inverse iteration method is a basic approach for solving matrix eigenvalue problems. Now, two-grid methods based on the shifted inverse iteration have been established by Yang and Bi. Combining the correction technique proposed by Lin and Xie and the shifted inverse iteration, a multilevel correction scheme for the Steklov eigenvalue problem is proposed in this paper and the rigorous theoretical analysis is given. This type of multilevel method includes solving boundary value problems by the shifted inverse iteration in ?ner and ?ner ?nite element spaces,i.e., using the previously obtained eigenvalue as the shift and eigenfunction as the iteration initial value, and Steklov eigenvalue problems in coarse ?nite element spaces. And it is proved that the accuracy of the resulting eigenpair approximation can be improved after each correction step. Finally, numerical experiments with Matlab on the unit square and L-shape domain are reported to validate the e?ciency of the proposed method. In addition, with the iteration times increasing, the approximate eigenvalues are closer to the exact ones which lead the equation more closer to singular, although it is not di?cult to solve numerically. However, we can ?x the shift in the one correction step and get a new scheme. We will report a corresponding numerical experiment. The theoretical analysis and numerical experiments indicate that the scheme proposed in this paper is e?cient for both simple and multiple eigenvalues of the Steklov eigenvalue problem.