The Multiscale Finite Element Method with Nonconforming Elements for Elliptic Homogenization Problems

Abstract

The multiscale finite element method was developed by Hou and Wu [J. Comput. Phys., 134 (1997), pp. 169–189] to capture the effect of microscales on macroscales for multiscale problems through modification of finite element basis functions. For second-order multiscale partial differential equations, continuous (conforming) finite elements have been considered so far. Efendiev, Hou, and Wu [SIAM J. Numer. Anal., 37 (2000), pp. 888–910] considered a nonconforming multiscale finite element method where nonconformity comes from an oversampling technique for reducing resonance errors. In this paper we study the multiscale finite element method in the context of nonconforming finite elements for the first time. When the oversampling technique is used, a double nonconformity arises: one from this technique and the other from nonconforming elements. An equivalent formulation recently introduced by Chen [Numer. Methods Partial Differential Equations, 22 (2006), pp. 317–360] (also see [Y. R. Efendiev, T. Hou, and V. Ginting, Commun. Math. Sci., 2 (2004), pp. 553–589]) for the multiscale finite element method, which utilizes standard basis functions of finite element spaces but modifies the bilinear (quadratic) form in the finite element formulation of the underlying multiscale problems, is employed in the present study. Nonlinear multiscale and random homogenization problems are also studied, and numerical experiments are presented.