Numerical homogenization and correctors for nonlinear elliptic equations
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In this paper we consider numerical homogenization and correctors for nonlinear elliptic equations. The numerical correctors are constructed for operators with homogeneous random coefficients. The construction employs two scales, one a physical scale and the other a numerical scale. A numerical homogenization technique is proposed and analyzed. This procedure is developed within finite element formulation. The convergence of the numerical procedure is presented for the case of general heterogeneities using G-convergence theory. The proposed numerical homogenization procedure for elliptic equations can be considered as a generalization of multiscale finite element methods to nonlinear equations. Using corrector results we construct an approximation of oscillatory solutions. Numerical examples are presented. 2004 Society for Industrial and Applied Mathematics.