The interior penalty virtual element method for the biharmonic problem

Zhao, Jikun; Mao, Shipeng; Zhang, Bei; Wang, Fei

doi:10.1090/mcom/3828

The interior penalty virtual element method for the biharmonic problem
HTML articles powered by AMS MathViewer

by Jikun Zhao, Shipeng Mao, Bei Zhang and Fei Wang HTML | PDF

Math. Comp. 92 (2023), 1543-1574 Request permission

Abstract:

In this paper, an interior penalty virtual element method (IPVEM) is developed for solving the biharmonic problem on polygonal meshes. By modifying the existing $H^2$-conforming virtual element, an $H^1$-nonconforming virtual element is obtained with the same degrees of freedom as the usual $H^1$-conforming virtual element, such that it locally has $H^2$-regularity on each polygon in meshes. To enforce the $C^1$ continuity of the solution, an interior penalty formulation is adopted. Hence, this new numerical scheme can be regarded as a combination of the virtual element space and discontinuous Galerkin scheme. Compared with the existing methods, this approach has some advantages in reducing the degree of freedom and capability of handling hanging nodes. The well-posedness and optimal convergence of the IPVEM are proven in a mesh-dependent norm. We also derive a sharp estimate of the condition number of the linear system associated with IPVEM. Some numerical results are presented to verify the optimal convergence of the IPVEM and the sharp estimate of the condition number of the discrete problem. Besides, in the numerical test, the IPVEM has a good performance in computational accuracy by contrast with the other VEMs solving the biharmonic problem.

References

P. F. Antonietti, G. Manzini, and M. Verani, The fully nonconforming virtual element method for biharmonic problems, Math. Models Methods Appl. Sci. 28 (2018), no. 2, 387–407. MR 3741104, DOI 10.1142/S0218202518500100
Blanca Ayuso de Dios, Konstantin Lipnikov, and Gianmarco Manzini, The nonconforming virtual element method, ESAIM Math. Model. Numer. Anal. 50 (2016), no. 3, 879–904. MR 3507277, DOI 10.1051/m2an/2015090
Ivo Babuška and Miloš Zlámal, Nonconforming elements in the finite element method with penalty, SIAM J. Numer. Anal. 10 (1973), 863–875. MR 345432, DOI 10.1137/0710071
Garth A. Baker, Finite element methods for elliptic equations using nonconforming elements, Math. Comp. 31 (1977), no. 137, 45–59. MR 431742, DOI 10.1090/S0025-5718-1977-0431742-5
L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini, and A. Russo, Basic principles of virtual element methods, Math. Models Methods Appl. Sci. 23 (2013), no. 1, 199–214. MR 2997471, DOI 10.1142/S0218202512500492
L. Beirão da Veiga, F. Brezzi, L. D. Marini, and A. Russo, $H(\text {div})$ and $H(\mathbf {curl})$-conforming virtual element methods, Numer. Math. 133 (2016), no. 2, 303–332. MR 3489088, DOI 10.1007/s00211-015-0746-1
L. B. da Veiga, F. Brezzi, L. D. Marini, and A. Russo, Serendipity nodal VEM spaces, Comput. Fluids 141 (2016), 2–12.
Lourenco Beirão da Veiga, Carlo Lovadina, and Giuseppe Vacca, Divergence free virtual elements for the Stokes problem on polygonal meshes, ESAIM Math. Model. Numer. Anal. 51 (2017), no. 2, 509–535. MR 3626409, DOI 10.1051/m2an/2016032
L. Beirão da Veiga, F. Dassi, and A. Russo, A $C^1$ virtual element method on polyhedral meshes, Comput. Math. Appl. 79 (2020), no. 7, 1936–1955. MR 4070333, DOI 10.1016/j.camwa.2019.06.019
Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 1994. MR 1278258, DOI 10.1007/978-1-4757-4338-8
Susanne C. Brenner and Li-Yeng Sung, $C^0$ interior penalty methods for fourth order elliptic boundary value problems on polygonal domains, J. Sci. Comput. 22/23 (2005), 83–118. MR 2142191, DOI 10.1007/s10915-004-4135-7
Susanne C. Brenner and Li-Yeng Sung, Virtual element methods on meshes with small edges or faces, Math. Models Methods Appl. Sci. 28 (2018), no. 7, 1291–1336. MR 3815658, DOI 10.1142/S0218202518500355
Susanne C. Brenner, Kening Wang, and Jie Zhao, Poincaré-Friedrichs inequalities for piecewise $H^2$ functions, Numer. Funct. Anal. Optim. 25 (2004), no. 5-6, 463–478. MR 2106270, DOI 10.1081/NFA-200042165
Franco Brezzi and L. Donatella Marini, Virtual element methods for plate bending problems, Comput. Methods Appl. Mech. Engrg. 253 (2013), 455–462. MR 3002804, DOI 10.1016/j.cma.2012.09.012
Andrea Cangiani, Gianmarco Manzini, and Oliver J. Sutton, Conforming and nonconforming virtual element methods for elliptic problems, IMA J. Numer. Anal. 37 (2017), no. 3, 1317–1354. MR 3671497, DOI 10.1093/imanum/drw036
Long Chen and Jianguo Huang, Some error analysis on virtual element methods, Calcolo 55 (2018), no. 1, Paper No. 5, 23. MR 3760899, DOI 10.1007/s10092-018-0249-4
Long Chen and Xuehai Huang, Nonconforming virtual element method for $2m$th order partial differential equations in $\Bbb {R}^n$, Math. Comp. 89 (2020), no. 324, 1711–1744. MR 4081916, DOI 10.1090/mcom/3498
Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 0520174
G. Engel, K. Garikipati, T. J. R. Hughes, M. G. Larson, L. Mazzei, and R. L. Taylor, Continuous/discontinuous finite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity, Comput. Methods Appl. Mech. Engrg. 191 (2002), no. 34, 3669–3750. MR 1915664, DOI 10.1016/S0045-7825(02)00286-4
P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 775683
Jianguo Huang, Xuehai Huang, and Weimin Han, A new $C^0$ discontinuous Galerkin method for Kirchhoff plates, Comput. Methods Appl. Mech. Engrg. 199 (2010), no. 23-24, 1446–1454. MR 2630154, DOI 10.1016/j.cma.2009.12.012
Jianguo Huang and Yue Yu, A medius error analysis for nonconforming virtual element methods for Poisson and biharmonic equations, J. Comput. Appl. Math. 386 (2021), Paper No. 113229, 20. MR 4163091, DOI 10.1016/j.cam.2020.113229
Jim Douglas Jr., Todd Dupont, Peter Percell, and Ridgway 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), no. 3, 227–255 (English, with French summary). MR 543934, DOI 10.1051/m2an/1979130302271
Shuang Li and Kening Wang, Condition number estimates for $C^0$ interior penalty methods, Domain decomposition methods in science and engineering XVI, Lect. Notes Comput. Sci. Eng., vol. 55, Springer, Berlin, 2007, pp. 675–682. MR 2334162, DOI 10.1007/978-3-540-34469-8_{8}4
L. S. D. Morley, The triangular equilibrium element in the solution of plate bending problems, Aero. Quart. 19 (1968), 149–169.
I. Mozolevski and P. R. Bösing, Sharp expressions for the stabilization parameters in symmetric interior-penalty discontinuous Galerkin finite element approximations of fourth-order elliptic problems, Comput. Methods Appl. Math. 7 (2007), no. 4, 365–375. MR 2396828, DOI 10.2478/cmam-2007-0022
Igor Mozolevski and Endre Süli, A priori error analysis for the $hp$-version of the discontinuous Galerkin finite element method for the biharmonic equation, Comput. Methods Appl. Math. 3 (2003), no. 4, 596–607. MR 2048235, DOI 10.2478/cmam-2003-0037
Igor Mozolevski, Endre Süli, and Paulo R. Bösing, $hp$-version a priori error analysis of interior penalty discontinuous Galerkin finite element approximations to the biharmonic equation, J. Sci. Comput. 30 (2007), no. 3, 465–491. MR 2295480, DOI 10.1007/s10915-006-9100-1
Endre Süli and Igor Mozolevski, $hp$-version interior penalty DGFEMs for the biharmonic equation, Comput. Methods Appl. Mech. Engrg. 196 (2007), no. 13-16, 1851–1863. MR 2298696, DOI 10.1016/j.cma.2006.06.014
Cameron Talischi, Glaucio H. Paulino, Anderson Pereira, and Ivan F. M. Menezes, PolyMesher: a general-purpose mesh generator for polygonal elements written in Matlab, Struct. Multidiscip. Optim. 45 (2012), no. 3, 309–328. MR 2897115, DOI 10.1007/s00158-011-0706-z
Huayi Wei, Xuehai Huang, and Ao Li, Piecewise divergence-free nonconforming virtual elements for Stokes problem in any dimensions, SIAM J. Numer. Anal. 59 (2021), no. 3, 1835–1856. MR 4279152, DOI 10.1137/20M1350479
Garth N. Wells and Nguyen Tien Dung, A $C^0$ discontinuous Galerkin formulation for Kirchhoff plates, Comput. Methods Appl. Mech. Engrg. 196 (2007), no. 35-36, 3370–3380. MR 2335272, DOI 10.1016/j.cma.2007.03.008
Jikun Zhao, Shaochun Chen, and Bei Zhang, The nonconforming virtual element method for plate bending problems, Math. Models Methods Appl. Sci. 26 (2016), no. 9, 1671–1687. MR 3529253, DOI 10.1142/S021820251650041X
Jikun Zhao and Bei Zhang, The curl-curl conforming virtual element method for the quad-curl problem, Math. Models Methods Appl. Sci. 31 (2021), no. 8, 1659–1690. MR 4307006, DOI 10.1142/S0218202521500354
Jikun Zhao, Bei Zhang, Shaochun Chen, and Shipeng Mao, The Morley-type virtual element for plate bending problems, J. Sci. Comput. 76 (2018), no. 1, 610–629. MR 3812981, DOI 10.1007/s10915-017-0632-3
Jikun Zhao, Bei Zhang, Shipeng Mao, and Shaochun Chen, The divergence-free nonconforming virtual element for the Stokes problem, SIAM J. Numer. Anal. 57 (2019), no. 6, 2730–2759. MR 4032861, DOI 10.1137/18M1200762