Trevor Leslie : Homogenization for Elliptic PDEs via Two-Scale Convergence

Posted by Trevor Leslie , part of the Graduate Analysis Seminar.

At

March 7, 2016, 2 p.m.

In

SEO 612

Abstract

We study the elliptic PDE $-\nabla \cdot (A^\varepsilon \nabla u^\varepsilon) = f$ on a bounded domain with Dirichlet boundary conditions, where $A^\varepsilon(x) = A(x/\varepsilon)$ and $A(y)$ is a periodic matrix with bounded coefficients. We derive an approximation of this equation of the form $-\nabla \cdot (\overline{A} \nabla u) = f$ and prove that the solution $u$ approximates the solution $u^\varepsilon$ of the original equation. The proof uses the method of two-scale convergence; basic properties of two-scale convergence will be stated (without proof) for completeness.