Convergence Analysis for The Numerical Boundary Corrector for Elliptic Equations with Rapidly Oscillating Coefficients

Abstract

We develop the convergence analysis of a numerical scheme for approximating the solution of the elliptic problem $L_{\epsilon}u_{\epsilon} =- \frac{\partial}{\partial x_{i}}a_{ij}(x/ \epsilon) \frac{\partial }{\partial x_{j}}u_{\epsilon}=f \mbox{in} \Omega, u_{\epsilon}=0 \mbox{on} \partial\Omega,$ where $a(y)=(a_{ij}(y))$ is a periodic symmetric positive definite matrix and $\Omega = (0,1)^2$. The major goal of the numerical scheme is to capture the $\epsilon$-scale of the oscillations of the solution $u_\epsilon$ on a mesh size $h>\epsilon (\mbox{or} h>>\epsilon)$. The numerical scheme is based on asymptotic expansions, constructive boundary corrector, and finite element approximations. New a priori error estimates are established for the asymptotic expansions and for the constructive boundary correctors under weak assumptions on the regularity of the problem. These estimates permit to establish sharp finite element error estimates and to consider composite materials applications. Depending on the regularity of the problem, we establish for the numerical scheme a priori error estimates of $O(h^2 + \epsilon^{3/2}+ \epsilon h )$ on the $L^2$-norm, and $O(h + \epsilon^{1+ \hat{\delta}})$ for the broken $H^1$-norm where $\hat{\delta} \in (-\frac{1}{4}, 0]$.