We construct a finite volume element scheme with a monotonicity correction for anisotropic diffusion problems on general quadrilateral meshes, where the strict convexity restriction on the meshes is removed. The main contributions of this paper include three aspects. Firstly, the classical finite volume element (C-FVE) method is extended to severely distorted quadrilateral meshes even with concave cells by virtue of a new overlapping dual partition and a new gradient approximation. In fact, the choice of this dual partition and gradient approximation is also conducive to the construction of a monotone scheme. Secondly, a new monotonicity correction is suggested, based on which we obtain a monotone finite volume element (M-FVE) method. The resulting M-FVE method still keeps the local conservation and is easy for implementation. Finally, we analyze theoretically the truncation error and the monotonicity for this scheme. Besides, the existence of a solution to this nonlinear scheme is proved by applying the Brouwer's fixed point theorem. Numerical results demonstrate that the M-FVE method has the approximate second-order accuracy and preserves well the positivity of the solution for both isotropic and anisotropic diffusion problems on severely distorted quadrilateral meshes.

针对一般四边形网格上的各向异性扩散问题，我们构建了具有单调性校正的有限体积单元方案，其中，对网格的严格凸度限制已消除。本文的主要贡献包括三个方面。首先，借助新的重叠对偶划分和新的梯度近似，将经典有限体积元（C-FVE）方法扩展到即使在具有凹单元的情况下严重变形的四边形网格。实际上，这种双重划分和梯度近似的选择也有利于单调方案的构造。其次，提出了一种新的单调性校正方法，在此基础上，我们得到了一种单调有限体积单元法（M-FVE）。最终的M-FVE方法仍然保留了局部保护，易于实现。最后，我们从理论上分析了该方案的截断误差和单调性。此外，通过应用布劳威尔不动点定理证明了该非线性方案的一个解。数值结果表明，M-FVE方法具有近似的二阶精度，并且很好地保留了在严重变形的四边形网格上对于各向同性和各向异性扩散问题的解的正性。