$\mathcal{P}_m$ Interior Penalty Nonconforming Finite Element Methods for $2m$-th Order PDEs in $\mathbb{R}^n$
Shuonan Wu, Jinchao Xu
TL;DR
The work develops interior-penalty nonconforming finite element methods for $2m$-th order PDEs on arbitrary-dimensional simplicial meshes, using minimal spaces $\mathcal{P}_m(T)$ to create weakly continuous, nonconforming elements. The penality is kept small: $\eta=\mathcal{O}(1)$, and the bilinear form contains only semi-definite jump terms, yielding a symmetric positive definite stiffness operator. The authors establish two complementary error frameworks: a Strang-type estimate under extra regularity yielding quasi-optimal rates, and a conforming-relatives approach that delivers similar guarantees without extra regularity. Numerical tests on smooth and singular domains confirm the theoretical predictions and demonstrate robustness for high-order problems in multiple dimensions.
Abstract
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.