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

2020 ◽  
Vol 28 (2) ◽  
pp. 911-933
Author(s):  
Jianguo Huang ◽  
◽  
Sen Lin
2016 ◽  
Vol 54 (1) ◽  
pp. 34-56 ◽  
Author(s):  
P. F. Antonietti ◽  
L. Beira͂o da Veiga ◽  
S. Scacchi ◽  
M. Verani

2019 ◽  
Vol 60 ◽  
pp. C201-C214
Author(s):  
Balaje Kalyanaraman ◽  
Bishnu Lamichhane ◽  
Michael Meylan

The virtual element method is an extension of the finite element method on polygonal meshes. The virtual element basis functions are generally unknown inside an element and suitable projections of the basis functions onto polynomial spaces are used to construct the elemental stiffness and mass matrices. We present a gradient recovery method based on an oblique projection, where the gradient of the L2-polynomial projection of a solution is projected onto a virtual element space. This results in a computationally efficient numerical method. We present numerical results computing the gradients on different polygonal meshes to demonstrate the flexibility of the method. References B. Ahmad, A. Alsaedi, F. Brezzi, L. D. Marini, and A. Russo. Equivalent projectors for virtual element methods. Comput. Math. Appl., 66(3):376391, 2013. doi:10.1016/j.camwa.2013.05.015. L. Beirao da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini, and A. Russo. Basic principles of virtual element methods. Math. Mod. Meth. Appl. Sci., 23(01): 199214, 2013. doi:10.1142/S0218202512500492. L. Beirao da Veiga, F. Brezzi, L. D. Marini, and A. Russo. The hitchhiker's guide to the virtual element method. Math. Mod. Meth. Appl. Sci., 24(08): 15411573, 2014. doi:10.1142/S021820251440003X. Ilyas, M. and Lamichhane, B. P. and Meylan, M. H. A gradient recovery method based on an oblique projection and boundary modification. In Proceedings of the 18th Biennial Computational Techniques and Applications Conference, CTAC-2016, volume 58 of ANZIAM J., pages C34C45, 2017. doi:10.21914/anziamj.v58i0.11730. B. P. Lamichhane. A gradient recovery operator based on an oblique projection. Electron. Trans. Numer. Anal., 37:166172, 2010. URL http://etna.mcs.kent.edu/volumes/2001-2010/vol37/abstract.php?vol=37&pages=166-172. O. J. Sutton. Virtual element methods. PhD thesis, University of Leicester, Department of Mathematics, 2017. URL http://hdl.handle.net/2381/39955. C. Talischi, G. H. Paulino, A. Pereira, and I. F. M. Menezes. Polymesher: a general-purpose mesh generator for polygonal elements written in matlab. Struct. Multidiscip. O., 45(3):309328, 2012. doi:10.1007/s00158-011-0706-z. G. Vacca and L. Beirao da Veiga. Virtual element methods for parabolic problems on polygonal meshes. Numer. Meth. Part. D. E., 31(6): 21102134, 2015. doi:10.1002/num.21982. J. Xu and Z. Zhang. Analysis of recovery type a posteriori error estimators for mildly structured grids. Math. Comput., 73:11391152, 2004. doi:10.1090/S0025-5718-03-01600-4.


Author(s):  
Juliette Chabassier ◽  
Sébastien Imperiale

In this work we present and analyse a time discretisation strategy for linear wave equations that aims at using locally in space the most adapted time discretisation among a family of implicit or explicit centered second order schemes. The proposed family of schemes is adapted to domain decomposition methods such as the mortar element method. They correspond in that case to local implicit schemes and to local time stepping. We show that, if some regularity properties of the solution are satisfied and if the time step verifies a stability condition, then the family of proposed time discretisations provides, in a strong norm, second order space-time convergence. Finally, we provide 1D and 2D numerical illustrations that confirm the obtained theoretical results and we compare our approach on 1D test cases to other existing local time stepping strategies for wave equations.


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