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

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...

[1]  Jinchao Xu.  Two-grid Discretization Techniques for Linear and Nonlinear PDEs , 1996 .

[2]  M. Freidlin,et al.  Random Perturbations of Dynamical Systems , 1984 .

[3]  Yuanle Ma,et al.  Computational methods for multiphase flows in porous media , 2007, Math. Comput..

[4]  A. Bensoussan,et al.  Asymptotic analysis for periodic structures , 1979 .

[5]  P. Raviart,et al.  Conforming and nonconforming finite element methods for solving the stationary Stokes equations I , 1973 .

[6]  E. Weinan,et al.  Analysis of the heterogeneous multiscale method for elliptic homogenization problems , 2004 .

[7]  ZHANGXIN CHEN,et al.  Analysis of the Multiscale Finite Element Method for Nonlinear and Random Homogenization Problems , 2007, SIAM J. Numer. Anal..

[8]  V. Zhikov,et al.  Homogenization of Differential Operators and Integral Functionals , 1994 .

[9]  Thomas Y. Hou,et al.  A Multiscale Finite Element Method for Elliptic Problems in Composite Materials and Porous Media , 1997 .

[10]  Zhangxin Chen,et al.  On the implementation of mixed methods as nonconforming methods for second-order elliptic problems , 1995 .

[11]  Zhangxin Chen,et al.  Multiscale methods for elliptic homogenization problems , 2006 .

[12]  J. Douglas,et al.  A Galerkin method for a nonlinear Dirichlet problem , 1975 .

[13]  Zhiming Chen,et al.  A mixed multiscale finite element method for elliptic problems with oscillating coefficients , 2003, Math. Comput..

[14]  A. H. Schatz,et al.  An observation concerning Ritz-Galerkin methods with indefinite bilinear forms , 1974 .

[15]  Andrey L. Piatnitski,et al.  Homogenization of random non stationary parabolic operators , 2006 .

[16]  Thomas Y. Hou,et al.  Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients , 1999, Math. Comput..

[17]  M. Avellaneda,et al.  Compactness methods in the theory of homogenization , 1987 .

[18]  T. Hou,et al.  Multiscale Finite Element Methods for Nonlinear Problems and Their Applications , 2004 .

[19]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[20]  Thomas Y. Hou,et al.  Convergence of a Nonconforming Multiscale Finite Element Method , 2000, SIAM J. Numer. Anal..