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Local refinement techniques for elliptic problems on cell-centered grids. I: Error analysis. (English) Zbl 0724.65093

Conservative approximations of divergence type second order elliptic boundary value problems on rectangular cell centered grids with local refinement are constructed. Three types of discretized fluxes with increasing accuracy are introduced: a simple symmetric expression, a more accurate, but nonsymmetric one and, finally, a (class of) improved symmetric formula.

For the resulting finite difference schemes uniqueness of solution and certain inequalities are proven. A priori error estimates in discrete ${H}^{1}$-norm are derived for the simple symmetric scheme and the improved schemes with order ${h}^{1/2}$, respectively ${h}^{3/2}$. The smoothness assumption for this result is that the solution of the continuous problem is in ${H}^{1+\alpha }$, $\alpha >1/2$ in the former, $\alpha >3/2$ in the latter case. The constant in the error estimate depends on the ratio of coarse to fine grid size, conclusions for a multilevel method are given.

For numerical results it is referred to a forthcoming paper.

##### MSC:
 65N06 Finite difference methods (BVP of PDE) 65N15 Error bounds (BVP of PDE) 35J25 Second order elliptic equations, boundary value problems