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Nonconforming box-schemes for elliptic problems on rectangular grids
Recently, Courbet and Croisille [Math.Model.Numer.Anal., 32, 631--649, 1998] introduced the FV box-scheme for the 2D Poisson problem in the case of triangular meshes. Generalization to higher degree box-schemes has been published by Croisille and Greff [Numer. Methods Partial Differential Equations, 18, 355--373, 2002].
These box-schemes are based on the idea of the finite volume method in that they take the average of the equations on each cell of the mesh. This gives rise to a natural choice of unknowns located at the interface of the mesh. Contrary to the finite volume method, these box-schemes are conservative and use only one mesh. They can be seen as a discrete mixed Petrov-Galerkin formulation of the Poisson problem. In this paper we focus our interest on box-schemes for the Poisson problem in 2D on rectangular grids. We discuss the basic FV box-scheme, and analyse and interpret it as three different box-schemes. The method is demonstrated by numerical examples.