Colorado State University

Variational Multiscale Methods for Computational Modeling of Heterogeneous Porous Media

By Todd Arbogast
From Department of Mathematics
University of Texas, Austin
When September 25, 2006
4:00 pm
Where Room WB202, Weber Building
Abstract If a coefficient in a differential equation is heterogeneous, i.e., it varies greatly on a small scale, then the solution will vary on that same small scale. An example of such a situation is flow in a natural porous medium, since normally the rock permeability varies on a small scale, and thus the fluid velocity changes greatly from point to point. This presents a computational challenge, since the numerical grid needs to resolve the finest scale, but this is much too detailed for even today's supercomputers to handle.

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.

Further Information James Liu
There will be Refreshments in WB 117 at 3.30pm
The Colloquium counts as Seminar Credit for Mathematics Students.