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June  2020, 28(2): 911-933. doi: 10.3934/era.2020048

A $ C^0P_2 $ time-stepping virtual element method for linear wave equations on polygonal meshes

School of Mathematical Science, and MOE-LSC, Shanghai Jiao Tong University, Shanghai 200240, China

* Corresponding author: jghuang@sjtu.edu.cn

Received  February 2020 Revised  April 2020 Published  June 2020

Fund Project: The work was partially supported by NSFC (Grant No. 11571237)

This paper is concerned with a $ C^0P_2 $ time-stepping virtual element method (VEM) for solving linear wave equations on polygonal meshes. The spatial discretization is carried out by the VEM while the temporal discretization is obtained based on the $ C^0P_2 $ time-stepping approach, leading to a fully discrete method. The error estimates in the $ H^1 $ semi-norm and $ L^2 $ norm are derived after detailed derivation. Finally, the numerical performance and efficiency of the proposed method is illustrated by several numerical experiments.

Citation: 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
References:
[1]

R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, Vol. 65. Academic Press, New York-London, 1975.  Google Scholar

[2]

B. AhmadA. AlsaediF. BrezziL. D. Marini and A. Russo, Equivalent projectors for virtual element methods, Comput. Math. Appl., 66 (2013), 376-391.  doi: 10.1016/j.camwa.2013.05.015.  Google Scholar

[3]

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

[4]

L. Beirão da VeigaF. Brezzi and L. D. Marini, Virtual elements for linear elasticity problems, SIAM J. Numer. Anal., 51 (2013), 794-812.  doi: 10.1137/120874746.  Google Scholar

[5]

L. Beirão da VeigaF. BrezziL. D. Marini and A. Russo, The hitchhiker's guide to the virtual element method, Math. Models Methods Appl. Sci., 24 (2014), 1541-1573.  doi: 10.1142/S021820251440003X.  Google Scholar

[6]

L. Beirão da VeigaF. BrezziL. D. Marini and A. Russo, Serendipity nodal VEM spaces, Comput. & Fluids, 141 (2016), 2-12.  doi: 10.1016/j.compfluid.2016.02.015.  Google Scholar

[7]

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

[8]

L. Beirão da VeigaF. Dassi and A. Russo, High-order virtual element method on polyhedral meshes, Comput. Math. Appl., 74 (2017), 1110-1122.  doi: 10.1016/j.camwa.2017.03.021.  Google Scholar

[9]

L. Beirão da VeigaC. Lovadina and D. Mora, A virtual element method for elastic and inelastic problems on polytope meshes, Comput. Methods Appl. Mech. Engrg., 295 (2015), 327-346.  doi: 10.1016/j.cma.2015.07.013.  Google Scholar

[10]

S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Third edition, Texts in Applied Mathematics, 15. Springer, New York, 2008. doi: 10.1007/978-0-387-75934-0.  Google Scholar

[11]

F. Brezzi, The great beauty of VEMs, Proceedings of the International Congress of Mathematicians - Seoul 2014, Kyung Moon Sa, Seoul, 1 (2014), 217-234.   Google Scholar

[12]

F. BrezziA. Buffa and K. Lipnikov, Mimetic finite differences for elliptic problems, M2AN Math. Model. Numer. Anal., 43 (2009), 277-295.  doi: 10.1051/m2an:2008046.  Google Scholar

[13]

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

[14]

A. CangianiG. Manzini and O. J. Sutton, Conforming and nonconforming virtual element methods for elliptic problems, IMA J. Numer. Anal., 37 (2017), 1317-1354.  doi: 10.1093/imanum/drw036.  Google Scholar

[15]

C. Chen, Structure Theory of Superconvergence of Finite Elements, Hunan Science & Technology Press: Changsha(in Chinese), 2001. Google Scholar

[16]

L. Chen and J. G. Huang, Some error analysis on virtual element methods, Calcolo, 55 (2018), 23 pp. doi: 10.1007/s10092-018-0249-4.  Google Scholar

[17]

L. ChenH. Y. Wei and M. Wen, An interface-fitted mesh generator and virtual element methods for elliptic interface problems, J. Comput. Phys., 334 (2017), 327-348.  doi: 10.1016/j.jcp.2017.01.004.  Google Scholar

[18]

F. FengW. M. Han and J. G. Huang, Virtual element method for an elliptic hmivariational inequality with applications to contact mechanics, J. Sci. Comput., 81 (2019), 2388-2412.  doi: 10.1007/s10915-019-01090-2.  Google Scholar

[19]

F. FengW. M. Han and J. G. Huang, Virtual element methods for elliptic variational inequalities of the second kind, J. Sci. Comput., 80 (2019), 60-80.  doi: 10.1007/s10915-019-00929-y.  Google Scholar

[20]

A. L. GainC. Talischi and G. H. Paulino, On the virtual element method for three-dimensional linear elasticity problems on arbitrary polyhedral meshes, Comput. Methods Appl. Mech. Engrg., 282 (2014), 132-160.  doi: 10.1016/j.cma.2014.05.005.  Google Scholar

[21]

J. J. LaiJ. G. Huang and C. M. Chen, Vibration analysis of plane elasticity problems by the $C^0$-continuous time stepping finite element method, Appl. Numer. Math., 59 (2009), 905-919.  doi: 10.1016/j.apnum.2008.04.001.  Google Scholar

[22]

J. J. LaiJ. G. Huang and Z. C. Shi, Vibration analysis for elastic multi-beam structures by the $C^0$-continuous time-stepping finite element method, Int. J. Numer. Methods Biomed. Eng., 26 (2010), 205-233.  doi: 10.1002/cnm.1143.  Google Scholar

[23]

I. PerugiaP. Pietra and A. Russo, A plane wave virtual element method for the Helmholtz problem, ESAIM Math. Model. Numer. Anal., 50 (2016), 783-808.  doi: 10.1051/m2an/2015066.  Google Scholar

[24]

P.-A. Raviart and J.-M. Thomas, Introduction à L'analyse Numérique des Équations aux Dérivées Partielles, Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris, 1983.  Google Scholar

[25]

O. J. Sutton, The virtual element method in 50 lines of MATLAB, Numer. Algorithms, 75 (2017), 1141-1159.  doi: 10.1007/s11075-016-0235-3.  Google Scholar

[26]

C. TalischiG. H. PaulinoA. Pereira and I. F. M. Menezes., PolyMesher: A general-purpose mesh generator for polygonal elements written in Matlab, Struct. Multidiscip. Optim., 45 (2012), 309-328.  doi: 10.1007/s00158-011-0706-z.  Google Scholar

[27]

G. Vacca, Virtual element methods for hyperbolic problems on polygonal meshes, Comput. Math. Appl., 74 (2017), 882-898.  doi: 10.1016/j.camwa.2016.04.029.  Google Scholar

[28]

G. Vacca and L. Beirão da Veiga, Virtual element methods for parabolic problems on polygonal meshes, Numer. Methods Partial Differential Equations, 31 (2015), 2110-2134.  doi: 10.1002/num.21982.  Google Scholar

[29]

P. WriggersW. T. Rust and B. D. Reddy, A virtual element method for contact, Comput. Mech., 58 (2016), 1039-1050.  doi: 10.1007/s00466-016-1331-x.  Google Scholar

[30]

J. K. ZhaoS. C. Chen and B. Zhang, The nonconforming virtual element method for plate bending problems, Math. Models Methods Appl. Sci., 26 (2016), 1671-1687.  doi: 10.1142/S021820251650041X.  Google Scholar

[31]

J. K. ZhaoB. ZhangS. C. Chen and S. P. Mao, The Morley-type virtual element for plate bending problems, J. Sci. Comput., 76 (2018), 610-629.  doi: 10.1007/s10915-017-0632-3.  Google Scholar

show all references

References:
[1]

R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, Vol. 65. Academic Press, New York-London, 1975.  Google Scholar

[2]

B. AhmadA. AlsaediF. BrezziL. D. Marini and A. Russo, Equivalent projectors for virtual element methods, Comput. Math. Appl., 66 (2013), 376-391.  doi: 10.1016/j.camwa.2013.05.015.  Google Scholar

[3]

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

[4]

L. Beirão da VeigaF. Brezzi and L. D. Marini, Virtual elements for linear elasticity problems, SIAM J. Numer. Anal., 51 (2013), 794-812.  doi: 10.1137/120874746.  Google Scholar

[5]

L. Beirão da VeigaF. BrezziL. D. Marini and A. Russo, The hitchhiker's guide to the virtual element method, Math. Models Methods Appl. Sci., 24 (2014), 1541-1573.  doi: 10.1142/S021820251440003X.  Google Scholar

[6]

L. Beirão da VeigaF. BrezziL. D. Marini and A. Russo, Serendipity nodal VEM spaces, Comput. & Fluids, 141 (2016), 2-12.  doi: 10.1016/j.compfluid.2016.02.015.  Google Scholar

[7]

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

[8]

L. Beirão da VeigaF. Dassi and A. Russo, High-order virtual element method on polyhedral meshes, Comput. Math. Appl., 74 (2017), 1110-1122.  doi: 10.1016/j.camwa.2017.03.021.  Google Scholar

[9]

L. Beirão da VeigaC. Lovadina and D. Mora, A virtual element method for elastic and inelastic problems on polytope meshes, Comput. Methods Appl. Mech. Engrg., 295 (2015), 327-346.  doi: 10.1016/j.cma.2015.07.013.  Google Scholar

[10]

S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Third edition, Texts in Applied Mathematics, 15. Springer, New York, 2008. doi: 10.1007/978-0-387-75934-0.  Google Scholar

[11]

F. Brezzi, The great beauty of VEMs, Proceedings of the International Congress of Mathematicians - Seoul 2014, Kyung Moon Sa, Seoul, 1 (2014), 217-234.   Google Scholar

[12]

F. BrezziA. Buffa and K. Lipnikov, Mimetic finite differences for elliptic problems, M2AN Math. Model. Numer. Anal., 43 (2009), 277-295.  doi: 10.1051/m2an:2008046.  Google Scholar

[13]

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

[14]

A. CangianiG. Manzini and O. J. Sutton, Conforming and nonconforming virtual element methods for elliptic problems, IMA J. Numer. Anal., 37 (2017), 1317-1354.  doi: 10.1093/imanum/drw036.  Google Scholar

[15]

C. Chen, Structure Theory of Superconvergence of Finite Elements, Hunan Science & Technology Press: Changsha(in Chinese), 2001. Google Scholar

[16]

L. Chen and J. G. Huang, Some error analysis on virtual element methods, Calcolo, 55 (2018), 23 pp. doi: 10.1007/s10092-018-0249-4.  Google Scholar

[17]

L. ChenH. Y. Wei and M. Wen, An interface-fitted mesh generator and virtual element methods for elliptic interface problems, J. Comput. Phys., 334 (2017), 327-348.  doi: 10.1016/j.jcp.2017.01.004.  Google Scholar

[18]

F. FengW. M. Han and J. G. Huang, Virtual element method for an elliptic hmivariational inequality with applications to contact mechanics, J. Sci. Comput., 81 (2019), 2388-2412.  doi: 10.1007/s10915-019-01090-2.  Google Scholar

[19]

F. FengW. M. Han and J. G. Huang, Virtual element methods for elliptic variational inequalities of the second kind, J. Sci. Comput., 80 (2019), 60-80.  doi: 10.1007/s10915-019-00929-y.  Google Scholar

[20]

A. L. GainC. Talischi and G. H. Paulino, On the virtual element method for three-dimensional linear elasticity problems on arbitrary polyhedral meshes, Comput. Methods Appl. Mech. Engrg., 282 (2014), 132-160.  doi: 10.1016/j.cma.2014.05.005.  Google Scholar

[21]

J. J. LaiJ. G. Huang and C. M. Chen, Vibration analysis of plane elasticity problems by the $C^0$-continuous time stepping finite element method, Appl. Numer. Math., 59 (2009), 905-919.  doi: 10.1016/j.apnum.2008.04.001.  Google Scholar

[22]

J. J. LaiJ. G. Huang and Z. C. Shi, Vibration analysis for elastic multi-beam structures by the $C^0$-continuous time-stepping finite element method, Int. J. Numer. Methods Biomed. Eng., 26 (2010), 205-233.  doi: 10.1002/cnm.1143.  Google Scholar

[23]

I. PerugiaP. Pietra and A. Russo, A plane wave virtual element method for the Helmholtz problem, ESAIM Math. Model. Numer. Anal., 50 (2016), 783-808.  doi: 10.1051/m2an/2015066.  Google Scholar

[24]

P.-A. Raviart and J.-M. Thomas, Introduction à L'analyse Numérique des Équations aux Dérivées Partielles, Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris, 1983.  Google Scholar

[25]

O. J. Sutton, The virtual element method in 50 lines of MATLAB, Numer. Algorithms, 75 (2017), 1141-1159.  doi: 10.1007/s11075-016-0235-3.  Google Scholar

[26]

C. TalischiG. H. PaulinoA. Pereira and I. F. M. Menezes., PolyMesher: A general-purpose mesh generator for polygonal elements written in Matlab, Struct. Multidiscip. Optim., 45 (2012), 309-328.  doi: 10.1007/s00158-011-0706-z.  Google Scholar

[27]

G. Vacca, Virtual element methods for hyperbolic problems on polygonal meshes, Comput. Math. Appl., 74 (2017), 882-898.  doi: 10.1016/j.camwa.2016.04.029.  Google Scholar

[28]

G. Vacca and L. Beirão da Veiga, Virtual element methods for parabolic problems on polygonal meshes, Numer. Methods Partial Differential Equations, 31 (2015), 2110-2134.  doi: 10.1002/num.21982.  Google Scholar

[29]

P. WriggersW. T. Rust and B. D. Reddy, A virtual element method for contact, Comput. Mech., 58 (2016), 1039-1050.  doi: 10.1007/s00466-016-1331-x.  Google Scholar

[30]

J. K. ZhaoS. C. Chen and B. Zhang, The nonconforming virtual element method for plate bending problems, Math. Models Methods Appl. Sci., 26 (2016), 1671-1687.  doi: 10.1142/S021820251650041X.  Google Scholar

[31]

J. K. ZhaoB. ZhangS. C. Chen and S. P. Mao, The Morley-type virtual element for plate bending problems, J. Sci. Comput., 76 (2018), 610-629.  doi: 10.1007/s10915-017-0632-3.  Google Scholar

Figure 1.  Degrees of freedom for $ k = 1, 2, 3 $. Denote $ \chi_v $ with black dots, $ \chi_e^{k-2} $ with red squares, and $ \chi_E^{k-2} $ with blue diamonds
Figure 2.  The orders of errors $ E_{H^1} $ and $ Et_{L^2} $ in space direction for $ k = 1 $.
Figure 3.  The orders of errors $ E_{H^1} $ and $ Et_{L^2} $ in time direction for $ k = 1 $
Figure 6.  The orders of relative errors vs $ \# $Dofs for $ k = 1 $ and $ k = 2 $
Figure 4.  The orders of errors $ E_{H^1} $ and $ Et_{L^2} $ in space direction for $ k = 2 $
Figure 5.  The orders of errors $ E_{H^1} $ and $ Et_{L^2} $ in time direction for $ k = 2 $
Table 1.  $ E_{H^1},Et_{L^2} $ vs $ h $: fixed $ \tau = 1/40 $ and $ h $ varies from $ 1/5 $ to $ 1/80 $ for $ k = 1 $.
$ h $1/51/101/201/401/80
$ E_{H^1} $4.838e-22.438e-21.266e-25.582e-32.968e-3
$ Et_{L^2} $1.866e-24.734e-31.191e-32.837e-47.381e-5
$ h $1/51/101/201/401/80
$ E_{H^1} $4.838e-22.438e-21.266e-25.582e-32.968e-3
$ Et_{L^2} $1.866e-24.734e-31.191e-32.837e-47.381e-5
Table 2.  $ E_{H^1} $ vs $ \tau $: $ \tau $ varies from $ 1/2 $ to $ 1/6 $ with $ h = {\tau}^3 $ for $ k = 1 $.
$ \tau $ 1/2 1/3 1/4 1/5 1/6
$ Et_{H^1} $ 5.435e-2 1.703e-2 7.083e-3 3.622e-3 2.073e-3
$ \tau $ 1/2 1/3 1/4 1/5 1/6
$ Et_{H^1} $ 5.435e-2 1.703e-2 7.083e-3 3.622e-3 2.073e-3
Table 3.  $ Et_{L^2} $ vs $ \tau $: $ \tau $ varies from $ 1/4 $ to $ 1/16 $ with $ h^2 = {\tau}^3 $ for $ k = 1 $.
$ \tau $ 1/4 1/6 1/8 1/10 1/12 1/14 1/16
$ Et_{L^2} $ 9.738e-3 3.190e-3 1.350e-3 6.932e-4 3.955e-4 2.501e-4 1.647e-4
$ \tau $ 1/4 1/6 1/8 1/10 1/12 1/14 1/16
$ Et_{L^2} $ 9.738e-3 3.190e-3 1.350e-3 6.932e-4 3.955e-4 2.501e-4 1.647e-4
Table 4.  Error results with different meshes for $ k = 1 $.
$ h=\tau $ $ \# $Dofs $ E_{H^1} $ $ \frac{E_{H^1}}{h+{\tau}^3} $ $ \frac{E_{H^1}}{|u|_{1,h}} $ $ Et_{L^2} $ $ \frac{Et_{L^2}}{h^2+{\tau}^3} $ $ \frac{Et_{L^2}}{\|u\|_{0,h}} $
1/4 34 6.731e-2 0.2534 3.770e-2 4.215e-2 0.5395 6.897e-2
1/8 130 2.823e-2 0.2227 1.528e-2 7.310e-3 0.4158 1.172e-2
1/16 510 1.328e-2 0.2116 7.121e-3 1.832e-3 0.4413 2.867e-3
1/32 2047 6.990e-3 0.2235 3.742e-3 4.507e-4 0.4476 7.031e-4
1/64 8160 3.464e-3 0.2216 1.853e-3 1.069e-4 0.4310 1.666e-4
1/128 32630 1.767e-3 0.2262 9.455e-4 2.666e-5 0.4333 4.154e-5
$ h=\tau $ $ \# $Dofs $ E_{H^1} $ $ \frac{E_{H^1}}{h+{\tau}^3} $ $ \frac{E_{H^1}}{|u|_{1,h}} $ $ Et_{L^2} $ $ \frac{Et_{L^2}}{h^2+{\tau}^3} $ $ \frac{Et_{L^2}}{\|u\|_{0,h}} $
1/4 34 6.731e-2 0.2534 3.770e-2 4.215e-2 0.5395 6.897e-2
1/8 130 2.823e-2 0.2227 1.528e-2 7.310e-3 0.4158 1.172e-2
1/16 510 1.328e-2 0.2116 7.121e-3 1.832e-3 0.4413 2.867e-3
1/32 2047 6.990e-3 0.2235 3.742e-3 4.507e-4 0.4476 7.031e-4
1/64 8160 3.464e-3 0.2216 1.853e-3 1.069e-4 0.4310 1.666e-4
1/128 32630 1.767e-3 0.2262 9.455e-4 2.666e-5 0.4333 4.154e-5
Table 5.  $ E_{H^1},Et_{L^2} $ vs $ h $: fixed $ \tau = 1/80 $ and $ h $ varies from $ 1/5 $ to $ 1/80 $ for $ k = 2 $.
$ h $ 1/5 1/10 1/20 1/40 1/80
$ E_{H^1} $ 1.240e-2 1.994e-3 4.234e-4 9.892e-5 2.496e-5
$ Et_{L^2} $ 8.610e-3 3.312e-4 3.158e-5 3.585e-6 6.723e-7
$ h $ 1/5 1/10 1/20 1/40 1/80
$ E_{H^1} $ 1.240e-2 1.994e-3 4.234e-4 9.892e-5 2.496e-5
$ Et_{L^2} $ 8.610e-3 3.312e-4 3.158e-5 3.585e-6 6.723e-7
Table 6.  $ E_{H^1} $ vs $ \tau $: $ \tau $ varies from $ 1/4 $ to $ 1/16 $ with $ h^2 = {\tau}^3 $ for $ k = 2 $.
$ \tau $ 1/4 1/6 1/8 1/10 1/12 1/14 1/16
$ E_{H^1} $ 7.333e-3 1.990e-3 8.194e-4 4.040e-4 2.322e-4 1.449e-4 9.702e-5
$ \tau $ 1/4 1/6 1/8 1/10 1/12 1/14 1/16
$ E_{H^1} $ 7.333e-3 1.990e-3 8.194e-4 4.040e-4 2.322e-4 1.449e-4 9.702e-5
Table 7.  $ Et_{L^2} $ vs $ \tau $: $ \tau $ varies from $ 1/4 $ to $ 1/128 $ with $ h = {\tau} $ for $ k = 2 $.
$ \tau $ 1/4 1/8 1/16 1/32 1/64 1/128
$ Et_{L^2} $ 1.970e-21.021e-39.013e-51.056e-51.319e-61.631e-7
$ \tau $ 1/4 1/8 1/16 1/32 1/64 1/128
$ Et_{L^2} $ 1.970e-21.021e-39.013e-51.056e-51.319e-61.631e-7
Table 8.  Error results with different meshes for $ k = 2 $.
$h=\tau$#Dofs $E_{H^1}$ $\frac{E_{H^1}}{h+{\tau}^3}$ $\frac{E_{H^1}}{|u|_{1,h}}$ $Et_{L^2}$ $\frac{Et_{L^2}}{h^2+{\tau}^3}$ $\frac{Et_{L^2}}{\|u\|_{0,h}}$
1/4346.731e-20.25343.770e-24.215e-20.53956.897e-2
1/81302.823e-20.22271.528e-27.310e-30.41581.172e-2
1/165101.328e-20.21167.121e-31.832e-30.44132.867e-3
1/3220476.990e-30.22353.742e-34.507e-40.44767.031e-4
1/6481603.464e-30.22161.853e-31.069e-40.43101.666e-4
1/128326301.767e-30.22629.455e-42.666e-50.43334.154e-5
$h=\tau$#Dofs $E_{H^1}$ $\frac{E_{H^1}}{h+{\tau}^3}$ $\frac{E_{H^1}}{|u|_{1,h}}$ $Et_{L^2}$ $\frac{Et_{L^2}}{h^2+{\tau}^3}$ $\frac{Et_{L^2}}{\|u\|_{0,h}}$
1/4346.731e-20.25343.770e-24.215e-20.53956.897e-2
1/81302.823e-20.22271.528e-27.310e-30.41581.172e-2
1/165101.328e-20.21167.121e-31.832e-30.44132.867e-3
1/3220476.990e-30.22353.742e-34.507e-40.44767.031e-4
1/6481603.464e-30.22161.853e-31.069e-40.43101.666e-4
1/128326301.767e-30.22629.455e-42.666e-50.43334.154e-5
Table 9.  The comparison of $ E_{L^2} $: method (3.2) vs the Newmark trapezoidal VEM.
$\tau=h$ $C^0P_2$ time-stepping VEMNewmark trapezoidal VEM
$k$ $h$ #Dofs $E_{L^2}$ $\frac{E_{L^2}}{h^{k+1}+{\tau}^3}$ $\frac{E_{L^2}}{\| u\|_{0,h}}$ $E_{L^2}$ $\frac{E_{L^2}}{h^{k+1}+{\tau}^2}$ $\frac{E_{L^2}}{\| u\|_{0,h}}$
1/5501.119e-32.331e-23.480e-21.167e-31.458e-23.629e-2
1/102022.754e-42.504e-28.354e-32.806e-41.403e-28.512e-3
11/208017.751e-52.953e-22.332e-37.786e-51.557e-22.343e-3
1/4031892.088e-53.259e-26.269e-42.099e-51.679e-26.301e-4
1/80127605.097e-63.222e-21.529e-45.115e-61.637e-21.535e-4
1/51492.136e-41.335e-26.405e-32.267e-44.723e-36.798e-3
1/106032.074e-51.037e-26.222e-42.211e-52.010e-36.632e-4
21/2024012.916e-61.166e-28.747e-53.096e-61.179e-39.287e-5
1/4095773.605e-71.154e-21.081e-54.927e-77.691e-41.478e-5
1/80383194.535e-81.161e-21.360e-61.152e-77.285e-43.458e-6
$\tau=h$ $C^0P_2$ time-stepping VEMNewmark trapezoidal VEM
$k$ $h$ #Dofs $E_{L^2}$ $\frac{E_{L^2}}{h^{k+1}+{\tau}^3}$ $\frac{E_{L^2}}{\| u\|_{0,h}}$ $E_{L^2}$ $\frac{E_{L^2}}{h^{k+1}+{\tau}^2}$ $\frac{E_{L^2}}{\| u\|_{0,h}}$
1/5501.119e-32.331e-23.480e-21.167e-31.458e-23.629e-2
1/102022.754e-42.504e-28.354e-32.806e-41.403e-28.512e-3
11/208017.751e-52.953e-22.332e-37.786e-51.557e-22.343e-3
1/4031892.088e-53.259e-26.269e-42.099e-51.679e-26.301e-4
1/80127605.097e-63.222e-21.529e-45.115e-61.637e-21.535e-4
1/51492.136e-41.335e-26.405e-32.267e-44.723e-36.798e-3
1/106032.074e-51.037e-26.222e-42.211e-52.010e-36.632e-4
21/2024012.916e-61.166e-28.747e-53.096e-61.179e-39.287e-5
1/4095773.605e-71.154e-21.081e-54.927e-77.691e-41.478e-5
1/80383194.535e-81.161e-21.360e-61.152e-77.285e-43.458e-6
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