A pressure robust staggered discontinuous Galerkin method for the Stokes equations

Abstract

In this paper we propose a pressure robust staggered discontinuous Galerkin method for the Stokes equations on general polygonal meshes by using piecewise constant approximations. We modify the right hand side of the body force in the discrete formulation by exploiting divergence preserving velocity reconstruction operator, which is the crux for pressure independent velocity error estimates. The optimal convergence for velocity gradient, velocity and pressure are proved. In addition, we are able to prove the superconvergence of velocity approximation by the incorporation of divergence preserving velocity reconstruction operator in the dual problem, which is also an important contribution of this paper. Finally, several numerical experiments are carried out to confirm the theoretical findings.

Pressure-robust staggered DG methods for the Navier-Stokes equations on general meshes

“…Thanks to the divergence-free property and the specially designed term for the nonlinear convective term, we are able to prove that the convergence estimates are independent of the pressure variable and the coefficient ν under a suitable assumption on the source term f . Unlike the existing works on polygonal meshes [10,27,39,53], we do not require velocity reconstruction, which can greatly reduce the computational complexity and ease the construction of the method.…”

“…Then, a high order staggered DG method for general secondorder elliptic problems is developed in [51], and it is applied to various physical problems arising from practical applications [55,51,36,52,47,50,54,48]. For a pressure-robust method, Zhao et al [53] proposed a lowest-order pressure-robust staggered DG method for the Stokes equations. They used a reconstruction operator based on an H(div) conforming function space on polygonal meshes.…”