A postprocessing technique for mixed finite element methods for the Cahn-Hilliard equation is developed and analyzed. Once the mixed finite element approximations have been computed at a fixed time on the coarser mesh, the approximations are postprocessed by solving two decoupled Poisson equations in an enriched finite element space (either on a finer grid or a higher-order space) for which many fast Poisson solvers can be applied. The nonlinear iteration is only applied to a much smaller size problem and the computational cost using Newton and direct solvers is negligible compared with the cost of the linear problem. The analysis presented here shows that this technique remains the optimal rate of convergence for both the concentration and the chemical potential approximations. The corresponding error estimate obtained in our paper, especially the negative norm error estimates, are non-trivial and different with the existing results in the literatures.

中文翻译：

---

解决卡恩-希尔德方程的后处理混合有限元方法：方法和误差分析。

开发并分析了Cahn-Hilliard方程的混合有限元方法的后处理技术。一旦在固定时间在较粗的网格上计算了混合有限元逼近，就可以通过在富集有限元空间（在更精细的网格或更高阶的空间中）中求解两个解耦的泊松方程来对近似值进行后处理。可以应用快速的泊松求解器。非线性迭代仅适用于尺寸较小的问题，与线性问题的成本相比，使用牛顿法和直接求解器的计算成本可忽略不计。此处介绍的分析表明，对于浓度和化学势近似值，该技术均保持最佳收敛速度。在我们的论文中获得的相应误差估计，