Numerical boundary corrector methods and analysis for a second order elliptic PDE with highly oscillatory periodic coefficients with applications to porous media

Henrique Versieux

Keywords: Finite elements | homogenization | elliptic equations | multiscaling | boundary layer | mixed finite elements

We develop a numerical discretization for linear elliptic equations with rapidly oscillating coefficients. The major goal is to develop a numerical scheme on a mesh size $h>\epsilon$ (or $ h>>\epsilon$), capturing the solution oscillations occurring in a scale $\epsilon$. The proposed method is based on asymptotic expansion and a novel treatment on the boundary corrector term. We obtain discretization errors of $O(h^2 + \epsilon^{3/2}+ \epsilon h)$ and $O(h + \epsilon)$ for the $L^2$ norm and the broken semi-norm $H^1$, respectively. Numerical results are presented.

Anexos:

• versieux_tese.pdf