Abstract
We analyse the p- and hp-versions of the virtual element method (VEM) for the Stokes problem on polygonal domains. The key tool in the analysis is the existence of a bijection between Poisson-like and Stokes-like VE spaces for the velocities. This allows us to re-interpret the standard VEM for Stokes as a VEM, where the test and trial discrete velocities are sought in Poisson-like VE spaces. The upside of this fact is that we inherit from Beirão da Veiga et al. (Numer. Math. 138(3), 581–613, 2018) an explicit analysis of best interpolation results in VE spaces, as well as stabilization estimates that are explicit in terms of the degree of accuracy p of the method. We prove exponential convergence of the hp-VEM for Stokes problems with regular right-hand sides. We corroborate the theoretical estimates with numerical tests for both the p- and hp-versions of the method.
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Acknowledgements
The authors are very grateful to the anonymous referees for their valuable and constructive comments, which have contributed to the improvement of the paper. L. M. acknowledges support from the Austrian Science Fund (FWF) project P33477.
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Communicated by: Lourenco Beirao da Veiga
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Chernov, A., Marcati, C. & Mascotto, L. p- and hp- virtual elements for the Stokes problem. Adv Comput Math 47, 24 (2021). https://doi.org/10.1007/s10444-020-09831-w
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DOI: https://doi.org/10.1007/s10444-020-09831-w