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

TL;DR

This paper develops a unified framework for $H^m$-nonconforming virtual element methods applicable to any polytope in $\

Contribution

It introduces a generalized Green's identity and constructs $H^m$-nonconforming virtual elements of any order, enabling optimal approximation of $m$-harmonic equations in arbitrary polyhedral domains.

Findings

Established bounds on jump related to weak continuity

Proved optimal error estimates for the methods

Derived norm equivalence of stabilization term

Abstract

A unified construction of the $H^m$-nonconforming virtual elements of any order $k$ is developed on any shape of polytope in $\mathbb R^n$ with constraints $m\leq n$ and $k\geq m$. As a vital tool in the construction, a generalized Green's identity for $H^m$ inner product is derived. The $H^m$-nonconforming virtual element methods are then used to approximate solutions of the $m$-harmonic equation. After establishing a bound on the jump related to the weak continuity, the optimal error estimate of the canonical interpolation, and the norm equivalence of the stabilization term, the optimal error estimates are derived for the $H^m$-nonconforming virtual element methods.