In general $n$-dimensional simplicial meshes, we propose a family of interior penalty nonconforming finite element methods for $2m$-th order partial differential equations, where $m \geq 0$ and $n \geq 1$. For this family of nonconforming finite elements, the shape function space consists of polynomials with a degree not greater than $m$, making it minimal. This family of finite element spaces exhibits natural inclusion properties, analogous to those in the corresponding Sobolev spaces in the continuous case. By applying interior penalty to the bilinear form, we establish quasi-optimal error estimates in the energy norm. Due to the weak continuity of the nonconforming finite element spaces, the interior penalty terms in the bilinear form take a simple form, and an interesting property is that the penalty parameter needs only to be a positive constant of $\mathcal{O}(1)$. These theoretical results are further validated by numerical tests.

翻译：在一般的 $n$ 维单纯形网格上，我们提出了一族用于求解 $2m$ 阶偏微分方程的内罚非协调有限元方法，其中 $m \geq 0$ 且 $n \geq 1$。对于这族非协调有限元，其形函数空间由次数不超过 $m$ 的多项式构成，因此是最小的。这族有限元空间展现出自然的包含性质，类似于连续情形中相应 Sobolev 空间的性质。通过对双线性形式施加内罚，我们建立了能量范数下的拟最优误差估计。由于非协调有限元空间的弱连续性，双线性形式中的内罚项形式简单，并且一个有趣的性质是罚参数仅需为 $\mathcal{O}(1)$ 的正常数。这些理论结果通过数值试验得到了进一步验证。