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An Isaac Newton Institute Workshop

The Theory of Highly Oscillatory Problems

High dimensional finite element methods for elliptic problems with highly oscillating coefficients

29th March 2007

Author: Viet Ha Hoang (University of Cambridge)

Abstract

Elliptic homogenization problems in a domain $\Omega \subset \R^d$ with $n+1$ separated scales are reduced to elliptic one-scale problems in dimension $(n+1)d$. These one-scale problems are discretized by a sparse tensor product finite element method (FEM). We prove that this sparse FEM has accuracy, work and memory requirement comparable to standard FEM for single scale problems in $\Omega$ while it gives numerical approximations of the correct homogenized limit as well as of all first order correctors, throughout the physical domain with performance independent of the physical problem's scale parameters. Numerical examples for model diffusion problems with two and three scales confirm our results.