Consider the elliptic problem in , on , which is equivalent with: Find such that, for any admissible volume
Then, the finite volume element (FVE) method for approximating the solution (1) consists of defining a similar problem in a finite-dimensional subspace for a finite set of volumes , (,), for a given S: Find such that
We denote , the discretization error; , where u and w are the solutions of (1) and (2), respectively, and G is the composite grid, . A first evaluation error result is obtained by: If and , or 3, then:
An improved error is given by: If , where or 3, then:
where C is a constant independent of the mesh size ; ; ; are the Sobolev norm and seminorm, respectively, implicitly given in the paper. The paper contains a detailed presentation of the FVE method.