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

TL;DR

This paper develops a universal nonconforming virtual element method for high-order PDEs in any dimension, providing theoretical analysis and implementation details for solving the $m$-harmonic equation.

Contribution

It introduces a new construction of $H^m$-nonconforming virtual elements for $m>n$, including a generalized Green's identity and stability analysis.

Findings

Proved norm equivalence of stabilization on the kernel of the local $H^m$ projection.

Established optimal error estimates for the virtual element methods.

Discussed implementation strategies for the proposed methods.

Abstract

The $H^m$-nonconforming virtual elements of any order $k$ on any shape of polytope in $\mathbb R^n$ with constraints $m>n$ and $k\geq m$ are constructed in a universal way. A generalized Green's identity for $H^m$ inner product with $m>n$ is derived, which is essential to devise the $H^m$-nonconforming virtual elements. By means of the local $H^m$ projection and a stabilization term using only the boundary degrees of freedom, the $H^m$-nonconforming virtual element methods are proposed to approximate solutions of the $m$-harmonic equation. The norm equivalence of the stabilization on the kernel of the local $H^m$ projection is proved by using the bubble function technique, the Poincar\'e inquality and the trace inequality, which implies the well-posedness of the virtual element methods. The optimal error estimates for the $H^m$-nonconforming virtual element methods are achieved from an estimate of the weak continuity and the error estimate of the canonical interpolation. Finally, the implementation of the nonconforming virtual element method is discussed.