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 -finite volume element method. The new quadrature rule is a linear combination of trapezoidal and midpoint rules, and the weights depend on a parameter . The novelty of this work is that, for any fully anisotropic diffusion tensor, we provide some specific 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 can only involves the anisotropic diffusion tensor and the geometry of quadrilateral cell. An optimal error estimate is also proved on quasi-parallelogram meshes. Finally, the theoretical findings are validated by several numerical examples.

中文翻译：

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四边形网格上的一类新颖的 Q1 有限体积元方案


构造并分析了一系列新颖的等参双线性有限体积元格式，以解决一般凸四边形网格上的各向异性扩散问题。这些新格式是通过采用特殊的求积规则来近似经典有限体积元方法中的线积分而获得的。新的求积规则是梯形规则和中点规则的线性组合，权重取决于参数 。这项工作的新颖之处在于，对于任何完全各向异性扩散张量，我们提供了一些特定的方法来确保所提出的方案在任意平行四边形、准平行四边形、梯形和一些一般凸四边形网格上的矫顽力结果。更有趣的是，该参数只能涉及​​各向异性扩散张量和四边形单元的几何形状。还在准平行四边形网格上证明了最佳误差估计。最后，通过几个数值例子验证了理论结果。