Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

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
DOI: https://doi.org/10.1090/mcom/3498
Published electronically: December 26, 2019
Full-text PDF
View in AMS MathViewer New

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 65N30, 65N12, 65N22

Retrieve articles in all journals with MSC (2010): 65N30, 65N12, 65N22


Additional Information

Long Chen
Affiliation: Department of Mathematics, University of California at Irvine, Irvine, California 92697
Email: chenlong@math.uci.edu

Xuehai Huang
Affiliation: School of Mathematics, Shanghai University of Finance and Economics, Shanghai 200433, People’s Republic of China
Email: huang.xuehai@sufe.edu.cn

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