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
Abstract:
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.
MSC 2000:
65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
35B27 Homogenization; partial differential equations in media with periodic structure
