$\mathcal{P}_m$ 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 R n , for any m ≥ 0, n ≥ 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.

Geometric decompositions of the simplicial lattice and smooth finite elements in arbitrary dimension

“…Repeat the argument for each pair of edges to conclude {D(e, r 1 )\D(∆ 0 (e), r 0 ), e ∈ ∆ 1 (T )} are disjoint. Then (22) follows.…”

“…On the other side, the lowest order nonconforming finite elements on simplexes were devised in [21,20,23] for m ≤ n and k = m + 1. We refer to [22,11,12] for more H m -nonconforming finite elements and [4,13,14] for H m -conforming and nonconforming virtual elements on any shape of polytope K in R n .…”

A $C^{0}$ interior penalty method for $m$th-Laplace equation

“…The finite element space in [18] is generalized for m = d + 1 by Wu and Xu in [21]. Recently in [22], it is further generalized for arbitrary m and d but with stabilization along mesh interface in order to balance the weak continuity and the penalty terms. In order to obtain stability and optimal convergence in some discrete H m -norm, [18,21,22] propose to compute numerical approximation to D m u, such that their implementation may become quite complicated as m is large.…”

“…Recently in [22], it is further generalized for arbitrary m and d but with stabilization along mesh interface in order to balance the weak continuity and the penalty terms. In order to obtain stability and optimal convergence in some discrete H m -norm, [18,21,22] propose to compute numerical approximation to D m u, such that their implementation may become quite complicated as m is large. The finite element spaces in [3,13] can be used to solve numerically (1.1) with any source term f ∈ H −m (Ω).…”

Nonconforming Virtual Element Method for $2m$-th Order Partial Differential Equations in $\mathbb R^n$

“…For general m, nonconforming elements on the simplices are easier to construct than conforming ones. In [36,35], Wang and Xu constructed the minimal H m -nonconforming elements in any dimension with constraint m ≤ n. Recently Wu and Xu extended these minimal H m -nonconforming elements to m = n + 1 by enriching the space of shape functions with bubble functions in [39], and to arbitrary m and n by using the interior penalty technique in [38]. In two dimensions, Hu and Zhang designed the H mnonconforming elements on the triangle for any m in [30].…”