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

Authors: Long Chen and Xuehai Huang
Journal: Math. Comp. 89 (2020), 1711-1744
MSC (2010): Primary 65N30, 65N12, 65N22
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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Additional Information

Long Chen
Affiliation: Department of Mathematics, University of California at Irvine, Irvine, California 92697

Xuehai Huang
Affiliation: School of Mathematics, Shanghai University of Finance and Economics, Shanghai 200433, People’s Republic of China

Received by editor(s): November 7, 2018
Received by editor(s) in revised form: November 9, 2018, July 7, 2019, and September 9, 2019
Published electronically: December 26, 2019
Additional Notes: The first author was supported by NSF DMS-1913080
The second author was supported by the National Natural Science Foundation of China Project 11771338, the Fundamental Research Funds for the Central Universities 2019110066, and Zhejiang Provincial Natural Science Foundation of China Project LY17A010010
Article copyright: © Copyright 2019 American Mathematical Society