Numerical homogenization of non-linear diffusion problems on adaptive meshes

We propose an efficient numerical strategy for solving non-linear diffusion problems defined in a porous medium with highly oscillatory characteristics. This scheme is based on the classical homogenization theory and uses a locally mass-conservative formulation at different scales. In addition, we discuss some properties of the proposed non-linear solvers and use an error indicator to perform a local mesh refinement. The main idea is to compute the effective parameters in such a way that the computational complexity is reduced but preserving the accuracy. We illustrate the behaviour of the homogenization scheme and of the non-linear solvers by performing two numerical tests. We consider both a quasi-periodic example and a problem involving strong heterogeneities in a non-periodic medium.