The nonconforming virtual element method for Oseen’s equation using a stream-function formulation

Abstract

We approximate the solution of the stream function formulation of the Oseen equations on general domains by designing a nonconforming Morley-type virtual element method. Under a suitable assumption on the continuous problem’s coefficients, the discrete scheme is well-posed. By introducing an enriching operator, we derive an a priori estimate of the error in a discrete H2 norm. By post-processing the discrete stream function, we compute the discrete velocity and vorticity fields. Furthermore, we recover an approximate pressure field by solving a Stokes-like problem in a nonconforming Crouzeix–Raviart-type virtual element space that is in a Stokes-complex relation with the Morley-type space of the virtual element approximation. Finally, we confirm our theoretical estimates by solving benchmark problems that include a convex and a nonconvex domain.