On A Priori Error Analysis of Fully Discrete Heterogeneous Multiscale FEM

Abstract

Heterogeneous multiscale methods have been introduced by E and Engquist [Commun. Math. Sci., 1 (2003), pp. 87--132] as a methodology for the numerical computation of problems with multiple scales. Analyses of the methods for various homogenization problems have been done by several authors. These results were obtained under the assumption that the microscopic models (the cell problems in the homogenization context) are analytically given. For numerical computations, these microscopic models have to be solved numerically. Therefore, it is important to analyze the error transmitted on the macroscale by discretizing the fine scale. We give in this paper H1 and L2 a priori estimates of the fully discrete heterogeneous multiscale finite element method. Numerical experiments confirm that the obtained a priori estimates are sharp.