p- and hp- virtual elements for the Stokes problem

• Published in: Advances in computational mathematics Vol. 47; no. 2
• Main Authors: Chernov, A., Marcati, C., Mascotto, L.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.04.2021; Springer Nature B.V
• Subjects: Computational mathematics; Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Interpolation; Mathematical and Computational Biology; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Visualization; 65N15; 65N12; Polygonal meshes; Stokes equation; Virtual element methods; and; 76D07; Galerkin methods; 65N30
• ISSN: 1019-7168; 1572-9044
• DOI: 10.1007/s10444-020-09831-w

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.