Two-scale, Locally Conservative Subgrid Upscaling for Elliptic Problems
Friday, January 11, 2002 - 11:00am - 12:00pm
We present a two-scale framework for approximating the solution of a second order elliptic problem in divergence form. The problem is viewed as a system of two first order equations with the divergence equation representing conservation of some quantity. We explicitly decompose the solution into coarse and fine scale parts. Moreover, the differential problem splits into the coupled system (1) a coarse-scale elliptic problem in divergence form, and (2) a fine scale problem localized in space. Solving the second problem for the fine scale part of the solution in terms of the coarse part, we obtain an operator mapping the coarse scale to the fine. Substituting this operator in the coarse problem results in an upscaled problem posed entirely of the coarse scale. Numerical approximation by a subgrid upscaling technique gives a computable algorithm. Since the fine scale is localized in space, an efficient algorithm results by using an influence function (numerical Greens function) technique to solve the fine subgrid-scale problems independently of the coarse-grid approximation. Moreover, the coarse-scale problem remains locally conservative. After correcting the coarse scale solution on the subgrid-scale, we obtain a fine scale representation of the solution. We show that the scheme is second order accurate. Numerical examples representing flow in a porous medium are presented to illustrate the effectiveness and applicability of the method.