A Finite Volume Method for the Laplace Equation on Almost Arbitrary Two-dimensional Grids

We present a finite volume method based on the integration of the Laplace equation on both the cells of a primal almost arbitrary two-dimensional mesh and those of a dual mesh obtained by joining the centers of the cells of the primal mesh. The key ingredient is the definition of discrete gradient and divergence operators verifying a discrete Green formula. This method generalizes an existing finite volume method that requires “Voronoi-type” meshes. We show the equivalence of this finite volume method with a non-conforming finite element method with basis functions being P 1 on the cells, generally called “diamond-cells”, of a third mesh. Under geometrical conditions on these diamondcells, we prove a first-order convergence both in the H0 norm and in the L 2 norm. Superconvergence results are obtained on certain types of homothetically refined grids. Finally, numerical experiments confirm these results and also show second-order convergence in the L norm on general grids. They also indicate that this method performs particularly well for the approximation of the gradient of the solution, and may be used on degenerating triangular grids. An example of application on nonconforming locally refined grids is given. Mathematics Subject Classification. 35J05, 35J25, 65N12, 65N15, 65N30. Received: April 26, 2004. Revised: July 7, 2005. Introduction In this paper, we consider a finite volume method for the approximation of the Laplace equation: −∆φ = f (1) on a bounded domain Ω, supplemented with adequate boundary conditions. Given a (primal) mesh covering Ω, finite volume methods for this type of equation may be classified into two main distinct categories: “vertexcentered” methods and “cell-centered” methods. Vertex-centered methods compute approximate values of φ at the vertices of the primal mesh by integrating Equation (1) on dual cells associated to the vertices of the primal mesh. On the opposite, cell-centered methods compute approximate values of φ at the centers of the cells of the primal mesh by integrating Equation (1) on the primal cells. For a review of these methods, we