\(\mathcal{P}_m\) Interior Penalty Nonconforming Finite Element Methods for \(2m\)-th Order PDEs in \(\mathbb{R}^n\)

Abstract

In this work, we propose a family of interior penalty nonconforming finite element methods for \(2m\)-th order partial differential equations in \(\mathbb{R}^n\), for any \(m \geq 0\), \(n \geq 1\). For the nonconforming finite element, the shape function space consists of all polynomials with a degree not greater than \(m\) and is hence minimal. This family of finite element spaces has some natural inclusion properties as in the corresponding Sobolev spaces in the continuous cases. By applying the interior penalty to the bilinear form, we establish quasi-optimal error estimates in the energy norm, provided that the corresponding conforming relatives exist. These theoretical results are further validated by the numerical tests.