Some estimates of virtual element methods for fourth order problems

Qingguang Guan; Qingguang Guan

doi:10.3934/era.2021074

Electronic Research Archive

2021, Volume 29, Issue 6: 4099-4118. doi: 10.3934/era.2021074

Previous Article Next Article

Special Issues

Some estimates of virtual element methods for fourth order problems

Qingguang Guan ^,

Department of Mathematics, Temple University, Philadelphia, PA 19122, USA

Received: 01 November 2020 Revised: 01 July 2021 Published: 22 September 2021
Primary: 65N30; Secondary: 35J40

In this paper, we employ the techniques developed for second order operators to obtain the new estimates of Virtual Element Method for fourth order operators. The analysis bases on elements with proper shape regularity. Estimates for projection and interpolation operators are derived. Also, the biharmonic problem is solved by Virtual Element Method, optimal error estimates were obtained. Our choice of the discrete form for the right hand side function relaxes the regularity requirement in previous work and the error estimates between exact solutions and the computable numerical solutions were proved.
- Virtual element method,
- shape regularity,
- fourth order,
- projection and interpolation operators,
- error estimates
Citation: Qingguang Guan. Some estimates of virtual element methods for fourth order problems[J]. Electronic Research Archive, 2021, 29(6): 4099-4118. doi: 10.3934/era.2021074

Related Papers:

Abstract

In this paper, we employ the techniques developed for second order operators to obtain the new estimates of Virtual Element Method for fourth order operators. The analysis bases on elements with proper shape regularity. Estimates for projection and interpolation operators are derived. Also, the biharmonic problem is solved by Virtual Element Method, optimal error estimates were obtained. Our choice of the discrete form for the right hand side function relaxes the regularity requirement in previous work and the error estimates between exact solutions and the computable numerical solutions were proved.

References

[1]	R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Vol. 140. Academic press, 2003.
[2]	Basic principles of virtual element methods. Math. Models Methods Appl. Sci. (2013) 23: 199-214.
[3]	Virtual element method for general second-order elliptic problems on polygonal meshes. Math. Models Methods Appl. Sci. (2016) 26: 729-750.
[4]	Mixed virtual element methods for general second order elliptic problems on polygonal meshes. ESAIM Math. Model. Numer. Anal. (2016) 50: 727-747.
[5]	L. Beirão da Veiga, F. Brezzi, L. D. Marini and A. Russo, Virtual element implementation for general elliptic equations, In Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations, Springer International Publishing, (2016), 39–71.
[6]	Estimation of linear functionals on Sobolev spaces with applications to Fourier transforms and spline interpolation. SIAM J. Numer. Anal. (1970) 7: 112-124.
[7]	Some estimates for virtual element methods. Comput. Methods Appl. Math. (2017) 17: 553-574.
[8]	S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods (Third Edition), Springer-Verlag, New York, 2008. doi: 10.1007/978-0-387-75934-0
[9]	Basic principles of mixed virtual element methods. ESAIM Math. Model. Numer. Anal. (2014) 48: 1227-1240.
[10]	Virtual Element Methods for plate bending problems. Comput. Methods Appl. Mech. Engrg. (2013) 253: 455-462.
[11]	Virtual element method for fourth order problems: $L^2$ estimates. Comput. Math. Appl. (2016) 72: 1959-1967.
[12]	A family of $C^1$ finite elements with optimal approximation properties for various Galerkin methods for 2nd and 4th order problems. RAIRO Anal. Numér. (1979) 13: 227-255.
[13]	P. Grisvard, Singularities in Boundary Value Problems, in: Recherches en Mathématiques Appliquées (Research in Applied Mathematics), vol. 22, Masson, Springer-Verlag, Paris, Berlin, 1992.
[14]	Q. Guan, M. Gunzburger and W. Zhao, Weak-Galerkin finite element methods for a second-order elliptic variational inequality, Comput. Methods Appl. Mech. Engrg., 337 (2018), 677–688. doi: 10.1016/j.cma.2018.04.006
[15]	L. Mu, J. Wang and X. Ye, Weak Galerkin finite element methods for the biharmonic equation on polytopal meshes, Numer. Methods Partial Differential Equations, 30 (2014), 1003–1029. doi: 10.1002/num.21855
[16]	J. Nečas, Direct Methods in the Theory of Elliptic Equations, Springer Science & Business Media, 2012. doi: 10.1007/978-3-642-10455-8
[17]	J. Wang and X. Ye, A weak Galerkin mixed finite element method for second order elliptic problems, Math. Comp., 83 (2014), 2101–2126. doi: 10.1090/S0025-5718-2014-02852-4

Reader Comments

Your name:*

Email:*
© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)