Multiscale asymptotic analysis and computations for Steklov eigenvalue problem in periodically perforated domain

Abstract

A second‐order asymptotic analysis method is developed for the Steklov eigenvalue problem in periodically perforated domain. By the two‐scale expansions of the eigenfunctions and eigenvalues, the first‐ and second‐order cell functions defined on the representative cell are obtained successively, the homogenized elliptic eigenvalue problem is formulated, and effective coefficients are derived. The first‐ and second‐order correctors of the eigenvalues are expressed in terms of the integrations of the cell functions and homogenized eigenfunctions. The error estimations of the expansions of eigenvalues are established, and the corresponding finite element algorithm is proposed. Numerical examples are carried out, and both qualitative and quantitative comparisons with the solutions by classical finite element computations are performed. It is demonstrated that this asymptotic model is effective to capture the local details of the eigenfunctions by considering the second‐order expansion terms, and the algorithm presented in this work is efficient to obtain the accurate spectral properties of the porous material at lower cost.