A pressure-robust virtual element method for the Stokes problem

Abstract In this paper, we introduce a pressure-robust virtual element method for the Stokes problem on convex polygonal meshes . The method is based on the lowest order virtual element, in which the degrees of freedom for the velocity are simply given by the evaluations of velocity at the mesh vertices and the average values of normal velocity across the mesh edges, and the pressure is approximated by piecewise constants. In the standard virtual element scheme, an average of nodal values of test function is used in the approximation of right hand side to obtain the computability and achieve optimal approximation. However, such standard scheme involves a pressure contribution in the velocity error. To achieve the pressure-independent velocity approximation, we define an H ( div ) -conforming velocity reconstruction operator for the velocity test function and propose the modified scheme by employing it in the approximation of right-hand-side source term assembling. Compared with the standard scheme, the stiffness matrix keeps unchanged and only the approximation of the right hand side changes. The error estimates for the velocity and pressure have been proved, which imply that the velocity error is independent of pressure. Numerical experiments are shown to validate the theoretical conclusions.

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