High dimensional finite element method for multiscale nonlinear monotone parabolic equations

Abstract

We develop in this paper an essentially optimal finite element (FE) method for solving locally periodic nonlinear monotone parabolic equations in a domain D⊂Rd that depend on n separable microscopic scales. For nonlinear multiscale equations, it is not possible to form the homogenized equation explicitly numerically. The method solves the multiscale homogenized equation which is obtained from multiscale convergence. This equation contains all the necessary information: the solution to the homogenized equation which approximates the solution to the multiscale equation macroscopically, and the scale interacting terms which provide the microscopic information. However, it is posed in a high dimensional tensorized domain. We develop the sparse tensor product FE method for this equation that uses an essentially optimal number of degrees of freedom to obtain an approximation for the solution within a prescribed accuracy. We then construct numerical correctors from the FE solution. In the two scale case, we derive a new homogenization error from which an explicit error for the numerical corrector is established: it is the sum of the FE error and the homogenization error. Numerical examples illustrate the theoretical results.