A new multiscale finite element method for high-contrast elliptic interface problems

We introduce a new multiscale finite element method which is able to accurately capture solutions of elliptic interface problems with high contrast coefficients by using only coarse quasiuniform meshes, and without resolving the interfaces. A typical application would be the modelling of flow in a porous medium containing a number of inclusions of low (or high) permeability embedded in a matrix of high (respectively low) permeability. Our method is H^1- conforming, with degrees of freedom at the nodes of a triangular mesh and requiring the solution of subgrid problems for the basis functions on elements which straddle the coefficient interface but which use standard linear approximation otherwise. A key point is the introduction of novel coefficientdependent boundary conditions for the subgrid problems. Under moderate assumptions, we prove that our methods have (optimal) convergence rate of O(h) in the energy norm and O(h^2) in the L_2 norm where h is the (coarse) mesh diameter and the hidden constants in these estimates are independent of the “contrast” (i.e. ratio of largest to smallest value) of the PDE coefficient. For standard elements the best estimate in the energy norm would be O(h^(1/2−ε)) with a hidden constant which in general depends on the contrast. The new interior boundary conditions depend not only on the contrast of the coefficients, but also on the angles of intersection of the interface with the element edges

A Generalized Multiscale Finite Element Method for the Brinkman Equation

In this paper we consider the numerical upscaling of the Brinkman equation in the presence of high-contrast permeability fields. We develop and analyze a robust and efficient Generalized Multiscale Finite Element Method (GMsFEM) for the Brinkman model. In the fine grid, we use mixed finite element method with the velocity and pressure being continuous piecewise quadratic and piecewise constant finite element spaces, respectively. Using the GMsFEM framework we construct suitable coarse-scale spaces for the velocity and pressure that yield a robust mixed GMsFEM. We develop a novel approach to construct a coarse approximation for the velocity snapshot space and a robust small offline space for the velocity space. The stability of the mixed GMsFEM and a priori error estimates are derived. A variety of two-dimensional numerical examples are presented to illustrate the effectiveness of the algorithm.Comment: 22 page