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 P1 on the cells,
generally called “diamond-cells”, of a third mesh.
Under geometrical conditions on these diamond-cells,
we prove a first-order convergence both in the $\xHone_0$ norm
and in the L² 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 L² 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 non-conforming locally refined grids is given.