Mixed Multiscale Methods for Heterogeneous Elliptic Problems

摘要

We consider a second order elliptic problem written in mixed form, i.e., as a system of two first order equations. Such problems arise in many contexts, including flow in porous media. The coefficient in the elliptic problem (the permeability of the porous medium) is assumed to be spatially heterogeneous. The emphasis here is on accurate approximation of the solution with respect to the scale of variation in this coefficient. Homogenization and upscaling techniques alone are generally inadequate for this problem. As an alternative, multiscale numerical methods have been developed. They can be viewed in one of three equivalent frameworks: as a Galerkin or finite element method with nonpolynomial basis functions, as a variational multiscale method with standard finite elements, or as a domain decomposition method with restricted degrees of freedom on the interfaces. We treat each case, and discuss the advantages of the approach for devising effective local multiscale methods. Included is recent work on methods that incorporate information from homogenization theory and effective domain decomposition methods.