Variational Multiscale Methods for Computational Modeling of Heterogeneous Porous Media

Standard finite element approximation of the solution on a coarse grid fails. An alternative is to solve the overall partial differential equation by incorporating the heterogeneity directly into the finite element basis using multiscale finite elements. These are defined by solving local, or subgrid, problems that resolve the fine-scale variation of the coefficient. In this way, one can improve the overall resolution of the finite element approximation. We present variational aspects of the method, called the Variational Multiscale Method, for second order elliptic partial differential equations in mixed form (i.e., written as a system of two first order equations). The coarse and fine scales are handled by introducing a novel expansion of the solution based on a Hilbert space direct sum decomposition. We present theoretical convergence results and computational examples which demonstrate the effectiveness of the method.