Nonconforming Virtual Element Method for 2𝑚th Order Partial Differential Equations in ℝⁿ

Chen, Long; Huang, Xuehai

doi:10.1090/mcom/3498

Nonconforming Virtual Element Method for $2m$th Order Partial Differential Equations in $\mathbb {R}^n$
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by Long Chen and Xuehai Huang;

Math. Comp. 89 (2020), 1711-1744

DOI: https://doi.org/10.1090/mcom/3498

Published electronically: December 26, 2019

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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.

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