A nonconforming finite element method (FEM) is developed and investigated for the coupled Poisson–Nernst–Planck (PNP) equations with low order element. Then, by use of the special properties of this element, that is, the interpolation operator is equivalent to its projection operator, and the consistency error estimate can reach order of O(h²) which is one order higher than that of its interpolation error estimates when the exact solution belongs to H³(Ω), the superclose estimates of order O(h²) and O(h² + τ) in the broken H¹‐norm are derived with new techniques for the semidiscrete scheme and backward Euler fully discrete scheme, respectively. Further, through employing interpolation postprocessing approach, the corresponding global superconvergence results are obtained. Finally, some numerical results are provided to confirm the theoretical analysis. It seems that our results have never been found in the existing literature. Here h and τ denote the mesh size and time step, respectively.

中文翻译：

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泊松－能斯特－普朗克方程组的非协调有限元方法

针对具有低阶元素的泊松－能斯特－普朗克（PNP）耦合方程，开发并研究了非协调有限元方法（FEM）。然后，通过使用此元素的特殊属性，即插值运算符等于其投影运算符，并且一致性误差估计可以达到O（h ²）的阶数，该阶数比其插值误差的阶数高一阶。估计精确解属于H ³（Ω）时，阶为O（h ²）和O（h ²  +  τ）的超闭合估计分别用半离散方案和后向欧拉完全离散方案的新技术推导了破碎的H ¹范式中的。此外，通过采用插值后处理方法，可以获得相应的全局超收敛结果。最后，提供一些数值结果以证实理论分析。似乎我们的结果从未在现有文献中找到。在此，h和τ分别表示网格大小和时间步长。