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Zbl 0724.65093
Ewing, R.E.; Lazarov, R.D.; Vassilevski, P.S.
Local refinement techniques for elliptic problems on cell-centered grids. I: Error analysis. (English)
[J] Math. Comput. 56, No.194, 437-461 (1991). ISSN 0025-5718; ISSN 1088-6842/e

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. \par For the resulting finite difference schemes uniqueness of solution and certain inequalities are proven. A priori error estimates in discrete $H\sp 1$-norm are derived for the simple symmetric scheme and the improved schemes with order $h\sp{1/2}$, respectively $h\sp{3/2}$. The smoothness assumption for this result is that the solution of the continuous problem is in $H\sp{1+\alpha}$, $\alpha >1/2$ 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. \par For numerical results it is referred to a forthcoming paper.
[K.Frischmuth (Rostock)]

MSC 2000:: *65N06 Finite difference methods (BVP of PDE)
65N15 Error bounds (BVP of PDE)
35J25 Second order elliptic equations, boundary value problems

Keywords: cell centered grid; balance equation; Conservative approximations; divergence type second order elliptic boundary value problems; local refinement; finite difference schemes; error estimates; multilevel method

Cited in: Zbl 0840.65124

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