Multiscale finite element methods for nonlinear problems and their applications. (English) Zbl 1083.65105

Efendiev, Yalchin R.; Hou, Thomas Yizhao; Ginting, Victor

Commun. Math. Sci. 2, No. 4, 553-589 (2004).

Summary: We propose a generalization of multiscale finite element methods (MsFEM) 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 oversampling technique greatly reduces the error. The application of MsFEM to porous media flows is considered. Finally, we describe further generalizations of MsFFM to nonlinear time-dependent equations and discuss the convergence of the method for various kinds of heterogeneities.

MSC:

65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N55	Multigrid methods; domain decomposition for boundary value problems involving PDEs
35J65	Nonlinear boundary value problems for linear elliptic equations
65N12	Stability and convergence of numerical methods for boundary value problems involving PDEs
35K55	Nonlinear parabolic equations
65M60	Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
76S05	Flows in porous media; filtration; seepage
76M10	Finite element methods applied to problems in fluid mechanics

Keywords:

upscaling; oversampling; nonlinear parabolic equations; convergence; nonlinear elliptic equations; Numerical examples; porous media flows; nonlinear time-dependent equations