Abstract: With special control volumes and finite volume element spaces,two symmetry-preserving finite volume element schemes for stationary diffusion problems are established on unstructured quadrilateral grids.Saturated order of error in L²-norm and H¹-norm for discrete solutions under quasi-uniform partition is demonstrated as diffusion coefficient is smooth.Numerical examples verify theoretical results.It shows strong adaptability of the scheme on distorted quadrilateral grids and for diffusion problems with non-smooth coefficient.The second scheme shows super-approximation for flux function at central point of element as grids are orthogonal.

Key words: unstructured quadrilateral grid, symmetry-preserving finite volume element scheme, diffusion equation, error estimate