doi: 10.3934/era.2021074
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Some estimates of virtual element methods for fourth order problems

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

* Corresponding author: Qingguang Guan

Received  November 2020 Revised  July 2021 Early access September 2021

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.

Citation: Qingguang Guan. Some estimates of virtual element methods for fourth order problems. Electronic Research Archive, doi: 10.3934/era.2021074
References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Vol. 140. Academic press, 2003.  Google Scholar

[2]

L. Beirão da VeigaF. BrezziA. CangianiG. ManziniL. D. Marini and A. Russo, Basic principles of virtual element methods, Math. Models Methods Appl. Sci., 23 (2013), 199-214.  doi: 10.1142/S0218202512500492.  Google Scholar

[3]

L. Beirão da VeigaF. BrezziL. D. Marini and A. Russo, Virtual element method for general second-order elliptic problems on polygonal meshes, Math. Models Methods Appl. Sci., 26 (2016), 729-750.  doi: 10.1142/S0218202516500160.  Google Scholar

[4]

L. Beirão da VeigaF. BrezziL. D. Marini and A. Russo, Mixed virtual element methods for general second order elliptic problems on polygonal meshes, ESAIM Math. Model. Numer. Anal., 50 (2016), 727-747.  doi: 10.1051/m2an/2015067.  Google Scholar

[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.  Google Scholar

[6]

J. H. Bramble and S. R. Hilbert, Estimation of linear functionals on Sobolev spaces with applications to Fourier transforms and spline interpolation, SIAM J. Numer. Anal., 7 (1970), 112-124.  doi: 10.1137/0707006.  Google Scholar

[7]

S. C. BrennerQ. Guan and L.-Y. Sung, Some estimates for virtual element methods, Comput. Methods Appl. Math., 17 (2017), 553-574.  doi: 10.1515/cmam-2017-0008.  Google Scholar

[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.  Google Scholar

[9]

F. BrezziR. S. Falk and L. D. Marini, Basic principles of mixed virtual element methods, ESAIM Math. Model. Numer. Anal., 48 (2014), 1227-1240.  doi: 10.1051/m2an/2013138.  Google Scholar

[10]

F. Brezzi and L. D. Marini, Virtual Element Methods for plate bending problems, Comput. Methods Appl. Mech. Engrg., 253 (2013), 455-462.  doi: 10.1016/j.cma.2012.09.012.  Google Scholar

[11]

C. Chinosi and L. D. Marini, Virtual element method for fourth order problems: $L^2$ estimates, Comput. Math. Appl., 72 (2016), 1959-1967.  doi: 10.1016/j.camwa.2016.02.001.  Google Scholar

[12]

J. Douglas JrT. DupontP. Percell and R. Scott, A family of $C^1$ finite elements with optimal approximation properties for various Galerkin methods for 2nd and 4th order problems, RAIRO Anal. Numér., 13 (1979), 227-255.  doi: 10.1051/m2an/1979130302271.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

J. Nečas, Direct Methods in the Theory of Elliptic Equations, Springer Science & Business Media, 2012. doi: 10.1007/978-3-642-10455-8.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Vol. 140. Academic press, 2003.  Google Scholar

[2]

L. Beirão da VeigaF. BrezziA. CangianiG. ManziniL. D. Marini and A. Russo, Basic principles of virtual element methods, Math. Models Methods Appl. Sci., 23 (2013), 199-214.  doi: 10.1142/S0218202512500492.  Google Scholar

[3]

L. Beirão da VeigaF. BrezziL. D. Marini and A. Russo, Virtual element method for general second-order elliptic problems on polygonal meshes, Math. Models Methods Appl. Sci., 26 (2016), 729-750.  doi: 10.1142/S0218202516500160.  Google Scholar

[4]

L. Beirão da VeigaF. BrezziL. D. Marini and A. Russo, Mixed virtual element methods for general second order elliptic problems on polygonal meshes, ESAIM Math. Model. Numer. Anal., 50 (2016), 727-747.  doi: 10.1051/m2an/2015067.  Google Scholar

[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.  Google Scholar

[6]

J. H. Bramble and S. R. Hilbert, Estimation of linear functionals on Sobolev spaces with applications to Fourier transforms and spline interpolation, SIAM J. Numer. Anal., 7 (1970), 112-124.  doi: 10.1137/0707006.  Google Scholar

[7]

S. C. BrennerQ. Guan and L.-Y. Sung, Some estimates for virtual element methods, Comput. Methods Appl. Math., 17 (2017), 553-574.  doi: 10.1515/cmam-2017-0008.  Google Scholar

[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.  Google Scholar

[9]

F. BrezziR. S. Falk and L. D. Marini, Basic principles of mixed virtual element methods, ESAIM Math. Model. Numer. Anal., 48 (2014), 1227-1240.  doi: 10.1051/m2an/2013138.  Google Scholar

[10]

F. Brezzi and L. D. Marini, Virtual Element Methods for plate bending problems, Comput. Methods Appl. Mech. Engrg., 253 (2013), 455-462.  doi: 10.1016/j.cma.2012.09.012.  Google Scholar

[11]

C. Chinosi and L. D. Marini, Virtual element method for fourth order problems: $L^2$ estimates, Comput. Math. Appl., 72 (2016), 1959-1967.  doi: 10.1016/j.camwa.2016.02.001.  Google Scholar

[12]

J. Douglas JrT. DupontP. Percell and R. Scott, A family of $C^1$ finite elements with optimal approximation properties for various Galerkin methods for 2nd and 4th order problems, RAIRO Anal. Numér., 13 (1979), 227-255.  doi: 10.1051/m2an/1979130302271.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

J. Nečas, Direct Methods in the Theory of Elliptic Equations, Springer Science & Business Media, 2012. doi: 10.1007/978-3-642-10455-8.  Google Scholar

[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.  Google Scholar

Figure 1.  A subdomain
Figure 2.  Local d.o.f. for the lowest-order element: $ k = 2, \ (r, s, m) = (3, 1, -2) $ (left), and next to the lowest element: $ k = 3, \ (r, s, m) = (3, 2, -1) $ (right)
Figure 3.  $ \mathbb{P}_3 $ macroelement
[1]

Liupeng Wang, Yunqing Huang. Error estimates for second-order SAV finite element method to phase field crystal model. Electronic Research Archive, 2021, 29 (1) : 1735-1752. doi: 10.3934/era.2020089

[2]

Benedict Geihe, Martin Rumpf. A posteriori error estimates for sequential laminates in shape optimization. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1377-1392. doi: 10.3934/dcdss.2016055

[3]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247

[4]

Lijuan Wang, Jun Zou. Error estimates of finite element methods for parameter identifications in elliptic and parabolic systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1641-1670. doi: 10.3934/dcdsb.2010.14.1641

[5]

Jie Shen, Xiaofeng Yang. Error estimates for finite element approximations of consistent splitting schemes for incompressible flows. Discrete & Continuous Dynamical Systems - B, 2007, 8 (3) : 663-676. doi: 10.3934/dcdsb.2007.8.663

[6]

Selim Esedoḡlu, Fadil Santosa. Error estimates for a bar code reconstruction method. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1889-1902. doi: 10.3934/dcdsb.2012.17.1889

[7]

Jianguo Huang, Sen Lin. A $ C^0P_2 $ time-stepping virtual element method for linear wave equations on polygonal meshes. Electronic Research Archive, 2020, 28 (2) : 911-933. doi: 10.3934/era.2020048

[8]

Petr Knobloch. Error estimates for a nonlinear local projection stabilization of transient convection--diffusion--reaction equations. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 901-911. doi: 10.3934/dcdss.2015.8.901

[9]

Craig Cowan, Pierpaolo Esposito, Nassif Ghoussoub. Regularity of extremal solutions in fourth order nonlinear eigenvalue problems on general domains. Discrete & Continuous Dynamical Systems, 2010, 28 (3) : 1033-1050. doi: 10.3934/dcds.2010.28.1033

[10]

Baishun Lai, Qing Luo. Regularity of the extremal solution for a fourth-order elliptic problem with singular nonlinearity. Discrete & Continuous Dynamical Systems, 2011, 30 (1) : 227-241. doi: 10.3934/dcds.2011.30.227

[11]

Dominik Hafemeyer, Florian Mannel, Ira Neitzel, Boris Vexler. Finite element error estimates for one-dimensional elliptic optimal control by BV-functions. Mathematical Control & Related Fields, 2020, 10 (2) : 333-363. doi: 10.3934/mcrf.2019041

[12]

Wolf-Jürgen Beyn, Raphael Kruse. Two-sided error estimates for the stochastic theta method. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 389-407. doi: 10.3934/dcdsb.2010.14.389

[13]

Marita Holtmannspötter, Arnd Rösch, Boris Vexler. A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system. Mathematical Control & Related Fields, 2021, 11 (3) : 601-624. doi: 10.3934/mcrf.2021014

[14]

Shuhao Cao. A simple virtual element-based flux recovery on quadtree. Electronic Research Archive, , () : -. doi: 10.3934/era.2021054

[15]

Maria-Magdalena Boureanu. Fourth-order problems with Leray-Lions type operators in variable exponent spaces. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 231-243. doi: 10.3934/dcdss.2019016

[16]

Roberto de A. Capistrano–Filho, Márcio Cavalcante, Fernando A. Gallego. Forcing operators on star graphs applied for the cubic fourth order Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021190

[17]

Peng Gao. Carleman estimates for forward and backward stochastic fourth order Schrödinger equations and their applications. Evolution Equations & Control Theory, 2018, 7 (3) : 465-499. doi: 10.3934/eect.2018023

[18]

Huafei Di, Yadong Shang, Jiali Yu. Existence and uniform decay estimates for the fourth order wave equation with nonlinear boundary damping and interior source. Electronic Research Archive, 2020, 28 (1) : 221-261. doi: 10.3934/era.2020015

[19]

Hiroyuki Hirayama, Mamoru Okamoto. Well-posedness and scattering for fourth order nonlinear Schrödinger type equations at the scaling critical regularity. Communications on Pure & Applied Analysis, 2016, 15 (3) : 831-851. doi: 10.3934/cpaa.2016.15.831

[20]

A. Aghajani, S. F. Mottaghi. Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (3) : 887-898. doi: 10.3934/cpaa.2018044

2020 Impact Factor: 1.833

Metrics

  • PDF downloads (43)
  • HTML views (55)
  • Cited by (0)

Other articles
by authors

[Back to Top]