Multiscale techniques for parabolic equations

TL;DR

This paper develops a multiscale finite element method for parabolic equations with complex diffusion coefficients, achieving optimal convergence rates independent of coefficient variations, and validates it through numerical experiments.

Contribution

It introduces a generalized finite element approach using local orthogonal decomposition for multiscale parabolic problems with nonsmooth data.

Findings

Optimal order convergence proven in $L_ ext{infty}(L_2)$-norm

Method's convergence rate depends only on contrast, not on coefficient variations

Numerical examples confirm theoretical results

Abstract

We use the local orthogonal decomposition technique to derive a generalized finite element method for linear and semilinear parabolic equations with spatial multiscale diffusion coefficient. We consider nonsmooth initial data and a backward Euler scheme for the temporal discretization. Optimal order convergence rate, depending only on the contrast, but not on the variations in the diffusion coefficient, is proven in the $L_\infty(L_2)$-norm. We present numerical examples, which confirm our theoretical findings.