We study a symmetric \(Q_1\) -finite volume element scheme for anisotropic diffusion problems on unstructured convex quadrilateral meshes. This scheme was originally suggested in [39] where the diffusion coefficient is a scalar and the analysis was done for Poisson equations on a uniform rectangular mesh. In this paper, we prove the symmetry and positive definiteness of the global bilinear form, by which the well-posedness of the scheme is established on an arbitrary convex quadrilateral mesh with any mesh size and a full anisotropic diffusion tensor. Under a weak mesh assumption, which covers the structured convex quadrilateral mesh, we present a simply analysis (different from the standard cell analysis approach) for the proof of \(L^2\) coercivity of the symmetric scheme. Further, by Taylor expansion, the optimal \(L^2\) and \(H^1\) error estimates are verified on the bisection quadrilateral mesh with a full anisotropic diffusion tensor. Consequently, we improve the theoretical results in [39]. Finally, several numerical examples are provided to validate the theoretical findings.
