p- and hp- virtual elements for the Stokes problem - Chernov, A.; Marcati, C.; Mascotto, L. (Advances in computational mathematics, 2021)
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.