Multiscale methods for elliptic homogenization problems

In this article we study two families of multiscale methods for numerically solving elliptic homogenization problems. The recently developed multiscale finite element method (Hou and Wu, J Comp Phys 134 (1997), 169-189) captures the effect of microscales on macroscales through modification of finite element basis functions. Here we reformulate this method that captures the same effect through modification of bilinear forms in the finite element formulation. This new formulation is a general approach that can handle a large variety of differential problems and numerical methods. It can be easily extended to nonlinear problems and mixed finite element methods, for example. The latter extension is carried out in this article. The recently introduced heterogeneous multiscale method (Engquist and Engquist, Comm Math Sci 1 (2003), 87-132) is designed for efficient numerical solution of problems with multiscales and multiphysics. In the second part of this article, we study this method in mixed form (we call it the mixed heterogeneous multiscale method). We present a detailed analysis for stability and convergence of this new method. Estimates are obtained for the error between the homogenized and numerical multiscale solutions. Strategies for retrieving the microstructural information from the numerical solution are provided and analyzed. Relationship between the multiscale finite element and heterogeneous multiscale methods is discussed. (c) 2005 Wiley Periodicals, Inc.