Collection of abstracts 13th GAMM-Seminar Kiel on Numerical Treatment of Multi-Scale Problems January 24th to 26th, 1997.

Friday, January 24th, 1997

A Multigrid Homogenization Method

For many applications the coefficients of the mathematical model incorporate strong variations on several length scales. A typical example is the diffusion coefficient in porous media. In general the multiscale asymptotic analysis for the determination of so called effective values for the modeling of approximate coarse scale problems is far to costly. Therefore simple averaging techniques like arithmetic, harmonic mean and renormalization procedures are used, which are computationally efficient, but inaccurate for coefficients with strong variations.

We present a method to approximately determine the effective diffusion coefficient on a coarse scale level. We apply the Galerkin approximation with operator- or matrix-dependent prolongations and Schur complement approximations. This leads to energy-dependent averaging procedures and to averaged equations which describe the large-scale behavior of the problem in discrete form.

Limitations of the usage of well known matrix-dependent prolongations in this approach are discussed. Techniques such as smoothing of the prolongation are given, which improve the presented method significantly.

We explain our modelling approach derived from multigrid, discuss its properties and compare it with other averaging techniques.