A multiscale rectangular element method for elliptic problems with entirely small periodic coefficients

Abstract

The purpose of this paper is to solve elliptic problems with entirely small periodic configuration by so-called multiscale finite element method. A special multiscale rectangular element space is constructed whose base functions consist of a standard bilinear conforming finite element base functions defined on a relatively coarse partition compared with small configuration parameter plus special bubble-like functions which include the small configuration information. Meanwhile the error of the multiscale finite element solution is analysed and the optimal error estimate is obtained. Finally, a multilevel additive Schwarz preconditioning method is presented for solving the discrete problem.