Multiscale Finite Element Methods for Nonlinear Problems and Their Applications

Multiscale Finite Element Methods for Nonlinear Problems and Their Applications. Efendiev, Y., Hou, T., & Ginting, V. Communications in Mathematical Sciences, 2(4):553-589, 2004.

Paper abstract bibtex

In this paper we propose a generalization of multiscale finite element methods (Ms-FEM) to nonlinear problems. We study the convergence of the proposed method for nonlinear elliptic equations and propose an oversampling technique. Numerical examples demonstrate that the over-sampling technique greatly reduces the error. The application of MsFEM to porous media flows is considered. Finally, we describe further generalizations of MsFEM to nonlinear time-dependent equations and discuss the convergence of the method for various kinds of heterogeneities.

@article {MR2119929,
    AUTHOR = {Efendiev, Y. and Hou, T. and Ginting, V.},
     TITLE = {Multiscale {F}inite {E}lement {M}ethods for {N}onlinear {P}roblems and
              {T}heir {A}pplications},
  JOURNAL = {Communications in Mathematical Sciences},
    VOLUME = {2},
      YEAR = {2004},
    NUMBER = {4},
     PAGES = {553-589},
      ISSN = {1539-6746},
   MRCLASS = {65N30},
  MRNUMBER = {2119929 (2005m:65265)},
MRREVIEWER = {Karsten Urban},
       URL = {http://projecteuclid.org/euclid.cms/1109885498},
ABSTRACT="In this paper we propose a generalization of multiscale finite element methods (Ms-FEM) to nonlinear problems. We study the convergence of the proposed method for nonlinear elliptic equations and propose an oversampling technique. Numerical examples demonstrate that the over-sampling technique greatly reduces the error. The application of MsFEM to porous media flows is considered. Finally, we describe further generalizations of MsFEM to nonlinear time-dependent equations and discuss the convergence of the method for various kinds of heterogeneities."
}

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