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p- and hp- virtual elements for the Stokes equation

by A. Chernov and C. Marcati and L. Mascotto

(Report number 2020-41)

Abstract
We analyse the \(p\)- and \(hp\)-versions of the virtual element method (VEM) for the the Stokes problem on a polygonal domain. 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, L., Chernov, A., Mascotto, L., Russo, A.: Exponential convergence of the hp virtual element method with corner singularity. 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 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.

Keywords: Stokes equation; virtual element methods; polygonal meshes; p- and hp- Galerkin methods

@Techreport{CMM20_914,
  author = {A. Chernov and C. Marcati and L. Mascotto},
  title = {p- and hp- virtual elements for the Stokes equation},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2020-41},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2020/2020-41.pdf },
  year = {2020}
}

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