Local refinement techniques for elliptic problems on cell-centered grids. I. Error analysis

Abstract

A finite difference technique on rectangular cell-centered grids with local refinement is proposed in order to derive discretizations of second-order elliptic equations of divergence type approximating the so-called balance equation. Error estimates in a discrete H 1 {H^1} -norm are derived of order h 1 / 2 {h^{1/2}} for a simple symmetric scheme, and of order h 3 / 2 {h^{3/2}} for both a nonsymmetric and a more accurate symmetric one, provided that the solution belongs to H 1 + α {H^{1 + \alpha }} for α > 1 2 \alpha > \frac {1}{2} and α > 3 2 \alpha > \frac {3}{2} , respectively.