For the two dimensional anisotropic diffusion problems, we postprocess the $k$-th order finite element solution to satisfy the local conservation law on a certain dual mesh. The postprocessing technique is independently implemented by solving a 3-by-3 and 4-by-4 local linear algebraic system on each triangular and quadrilateral element, respectively. For any full anisotropic diffusion tensors, and arbitrary triangular and trapezoidal meshes, the existence and uniqueness of the postprocessed solutions are proved theoretically. Consequently, some existing results are improved, and the traditional $h^{1+\gamma }$-parallelogram assumption is replaced with a more general one. The optimal $H^1$ and $L^2$ norm error estimates of the postprocessed solutions are also proved. Finally, some numerical experiments are presented to validate the theoretical findings.

对于二维各向异性扩散问题，我们对后处理 $ķ$满足一定双网格上的局部守恒定律的二阶有限元解。通过分别在每个三角形和四边形元素上求解3×3和4×4局部线性代数系统来独立实现后处理技术。对于任何完整的各向异性扩散张量，以及任意的三角形和梯形网格，从理论上证明了后处理解的存在性和唯一性。因此，现有的一些结果得到了改善，而传统的$H^{\mathrm{1个}+\gamma }$-平行四边形假设被更通用的假设所代替。最优的$H^{\mathrm{1个}}$ 和 ${\mathrm{大号}}^{\mathrm{2个}}$还证明了后处理解的范数误差估计。最后，提出了一些数值实验来验证理论结果。