Preprint A445/2006
Convergence Analysis for the Numerical Boundary Corrector for Elliptic Equations with Rapidly Oscillating Coefficients
Versieux, Henrique
Keywords: finite elements | homogenization | elliptic equations | multiscaling | boundary layer | mixed finite elements
We develop the convergence analysis for a numerical scheme proposed for approximating the solution of the elliptic problem \[ L_{\epsilon}u_{\epsilon} =- \frac{\partial}{\partial x_{i}}a_{ij}(x/ \epsilon) \frac{\partial }{\partial x_{j}}u_{\epsilon}=f ~~ \mbox{in}~~ \Omega, \hspace{.3cm} u_{\epsilon}=0 ~~ \mbox{on}~~ \partial\Omega, \] where the matrix $a(y)=(a_{ij}(y))$ is symmetric positive definite and periodic with period $Y$. The major goal is to develop a numerical scheme capturing the solution oscillations in the $\epsilon$ scale on a mesh size $h>\epsilon~(\mbox{or}~ h>>\epsilon)$. The proposed method is based on asymptotic analysis and on numerical treatments for the boundary corrector terms, and the convergence analysis is based on asymptotic expansion estimates and finite elements analysis. We obtain discretization errors of $O(h^2 + \epsilon^{3/2}+ \epsilon h )$ and $O(h + \epsilon)$ in the $L^2$ norm and the broken $H^1$ semi-norm, respectively.