In this work, for the two dimensional anisotropic diffusion problem, the classical $k$th order finite element solution is postprocessed to obtain a new finite volume element solution, such that the new solution satisfies the local conservation property on a certain dual mesh, and converges to the analytic solution with optimal rates. The postprocessing algorithm has a local nature and can be conducted element by element. The novelty of this paper is the introduction of a new bubble function, which enables us to prove the existence and uniqueness of the postprocessed solution on arbitrary triangular or convex quadrilateral meshes with full anisotropic diffusion tensor. The theoretical findings are also verified by some numerical results.

在这项工作中，对于二维各向异性扩散问题，经典的$ķ$对有限元解进行后处理，得到新的有限体积元解，使新解满足某对偶网格上的局部守恒性质，并以最优速率收敛到解析解。后处理算法具有局部性，可以逐个元素进行。本文的新颖之处在于引入了一个新的气泡函数，它使我们能够证明后处理解在具有全各向异性扩散张量的任意三角形或凸四边形网格上的存在性和唯一性。一些数值结果也验证了理论发现。