An analysis of the isoparametric bilinear finite volume element method by applying the Simpson rule to quadrilateral meshes

In this work, we construct and study a special isoparametric bilinear finite volume element scheme for solving anisotropic diffusion problems on general convex quadrilateral meshes. The new scheme is obtained by employing the Simpson rule to approximate the line integrals in the classical isoparametric bilinear finite volume element method. By using the cell analysis approach, we suggest a sufficient condition to ensure the coercivity of the new scheme. The sufficient condition has an analytic expression, which only involves the anisotropic diffusion tensor and the geometry of quadrilateral mesh. This yields that for any diffusion tensor and quadrilateral mesh, we can directly judge whether this sufficient condition is satisfied. Specifically, this condition covers the traditional $h^{1+\gamma }$-parallelogram and some trapezoidal meshes with any full anisotropic diffusion tensor. An optimal $H^1$ error estimate of the proposed scheme is also obtained for a quasi-parallelogram mesh. The theoretical results are verified by some numerical experiments.