A Posteriori Error Estimates for the Heterogeneous Multiscale Finite Element Method for Elliptic Homogenization Problems

Abstract

In this paper, we derive a new approach for the numerical analysis of the heterogeneous multiscale finite element method (HM-FEM) for elliptic homogenization problems. The HM-FEM was introduced in [W. E and B. Engquist, Commun. Math. Sci., 1 (2003), pp. 87--132], and a priori error estimates were derived in [W. E, P. Ming, and P. Zhang, J. Amer. Math. Soc., 18 (2005), pp. 121--156]. The main idea in our approach is to reformulate the heterogeneous multiscale method as a direct finite element discretization of the two scale homogenized equation. This approach puts the problem into a variational form and allows us in a straightforward way to derive a priori and a posteriori error estimates between the HM-FEM solution and the solution of the two scale homogenized equation. The derived a posteriori error estimate is a new result which cannot be obtained with the previous approaches. The theoretical results are verified in numerical experiments, and the usability of the a posteriori error estimate for local adaptive mesh refinement of the HM-FEM is shown.