Numerical Homogenization of Nonlinear Random Parabolic Operators

Abstract

In this paper we study the numerical homogenization of nonlinear random parabolic equations. This procedure is developed within a finite element framework. A careful choice of multiscale finite element bases and the global formulation of the problem on the coarse grid allow us to prove the convergence of the numerical method to the homogenized solution of the equation. The relation of the proposed numerical homogenization procedure to multiscale finite element methods is discussed. Within our numerical procedure one is able to approximate the gradients of the solutions. To show this feature of our method we develop numerical correctors that contain two scales, the numerical and the physical. Finally, we would like to note that our numerical homogenization procedure can be used for the general type of heterogeneities.