A novel family of Q1-finite volume element schemes on quadrilateral meshes

A novel family of isoparametric bilinear finite volume element schemes are constructed and analyzed to solve the anisotropic diffusion problems on general convex quadrilateral meshes. These new schemes are obtained by employing a special quadrature rule to approximate the line integrals in classical $Q_1$-finite volume element method. The new quadrature rule is a linear combination of trapezoidal and midpoint rules, and the weights depend on a parameter ${\omega }_K$. The novelty of this work is that, for any fully anisotropic diffusion tensor, we provide some specific ${\omega }_K$ to ensure the coercivity result of the proposed schemes on arbitrary parallelogram, quasi-parallelogram, trapezoidal and some general convex quadrilateral meshes. More interesting is that, the parameter ${\omega }_K$ can only involves the anisotropic diffusion tensor and the geometry of quadrilateral cell. An optimal $H^1$ error estimate is also proved on quasi-parallelogram meshes. Finally, the theoretical findings are validated by several numerical examples.