Elliptic Equations in High-Contrast Media and Applications

TL;DR

This paper reviews recent advances in approximating solutions to elliptic equations with high-contrast coefficients, focusing on asymptotic expansions, numerical methods, and applications in multiscale finite elements and elasticity.

Contribution

It introduces new asymptotic expansion techniques for high-contrast elliptic problems and demonstrates their application in multiscale finite element methods and elasticity.

Findings

Asymptotic expansions effectively approximate solutions in high-contrast media.

Finite Element Method accurately computes terms of the asymptotic expansion.

Convergence is established for the expansion in elasticity problems.

Abstract

In this manuscript we review some recent results about approximation of solutions of elliptic problems with high-contrast coefficients. In particular, we detail the derivation of asymptotic expansions for the solution in terms of the high-contrast of the coefficients and we consider some interesting applications. We use the Finite Element Method, which is applied in the numerical computation of terms of the asymptotic expansion. We also present an application to Multiscale Finite Elements, in particular, we numerically design approximation for the term $u_0$ with local harmonic characteristic functions. We also show the case of the linear elasticity problem, where we study the asymptotic problem with one inelastic inclusion and we provide the convergence for this expansion problem. Finally, we state some conclusions and final comments.