Abstract:

Fully resolving flow in highly heterogeneous porous media is often a computationally intractable task. Several upscaling and variational multiscale methods have been developed to address this problem. In this talk we discuss the relationship between mortar finite element methods and multiscale finite element methods. The latter represent the solution as a sum of a coarse scale and a fine scale (subgrid) component and require solving local fine scale problems to compute the multiscale basis functions. Mortar methods with coarse mortar spaces also resolve the solution on the fine scale in each subdomain, but impose continuity conditions on the coarse scale. The mortar formulation is more flexible than existing variational multiscale methods, since it allows for locally varying the interface degrees of freedom if higher resolution is needed. A domain decomposition algorithm reduces the multiscale algebraic system to a coarse scale interface problem. We study the accuracy and efficiency of the mortar multiscale approach in the context of mixed finite element and related methods. We also consider extensions of the above methods to stochastic partial differential equations in an effort to quantify the uncertainty in the model.