Vertex-centered finite volume schemes of any order over quadrilateral meshes for elliptic boundary value problems

A family of any order finite volume (FV) schemes over quadrilateral meshes is analyzed under the framework of Petrov–Galerkin method. By constructing a special mapping from the trial space to the test space, a unified proof for the inf–sup condition of any order FV schemes is provided under a weak condition that the underlying mesh is an h 1 + γ , γ > 0 parallelogram mesh. The optimal convergence rate of FV solutions is then obtained with well-known techniques.