Analysis of a multiscale method for nonlinear nonmonotone elliptic problems

Following the framework of the heterogeneous multiscale method, we present a numerical method for nonlinear elliptic homogenization problems. We briefly review the numerical, relying on an efficient coupling of macro and micro solvers, for linear problems. A fully discrete analysis is then given for nonlinear (nonmonotone) problems, optimal convergence rates in the H¹ and L² norms are derived and the uniqueness of the method is shown on sufficiently fine macro and micro meshes.

Numerical examples confirm the theoretical convergence rates and illustrate the performance and versatility of our approach.