The Stokes complex for Virtual Elements with application to Navier--Stokes flows
Provides theoretical foundation for Virtual Element discretizations of incompressible flows, but is incremental as it extends prior work by the same authors.
The paper establishes an exact Stokes complex structure for the Virtual Element Method in 2D Stokes and Navier-Stokes flows, proving computability of differential operators and testing a new convective discretization. Numerical tests validate the theory.
In the present paper, we investigate the underlying Stokes complex structure of the Virtual Element Method for Stokes and Navier--Stokes introduced in previous papers by the same authors, restricting our attention to the two dimensional case. We introduce a Virtual Element space $Φ_h \subset H^2(Ω)$ and prove that the triad $\{Φ_h, V_h, Q_h\}$ (with $V_h$ and $Q_h$ denoting the discrete velocity and pressure spaces) is an exact Stokes complex. Furthermore, we show the computability of the associated differential operators in terms of the adopted degrees of freedom and explore also a different discretization of the convective trilinear form. The theoretical findings are supported by numerical tests.