A hybrid reconstructed discontinuous Galerkin and continuous Galerkin finite element method for incompressible flows on unstructured grids

Abstract

A hybrid reconstructed discontinuous Galerkin and continuous Galerkin method based on an incremental pressure projection formulation, termed rDG(PnPm)+CG(Pn) in this paper, is developed for solving the unsteady incompressible Navier–Stokes equations on unstructured grids. In this method, a reconstructed discontinuous Galerkin method (rDG(PnPm)) is used to discretize the velocity and a standard continuous Galerkin method (CG(Pn)) is used to approximate the pressure. The rDG(PnPm)+CG(Pn) method is designed to increase the accuracy of the hybrid DG(Pn)+CG(Pn) method and yet still satisfy Ladyženskaja–Babuška–Brezzi (LBB) condition, thus avoiding the pressure checkerboard instability. An upwind method is used to discretize the nonlinear convective fluxes in the momentum equations in order to suppress spurious oscillations in the velocity field. A number of incompressible flow problems for a variety of flow conditions are computed to numerically assess the spatial order of convergence of the rDG(PnPm)+CG(Pn) method. The numerical experiments indicate that both rDG(P0P1)+CG(P1) and rDG(P1P2)+CG(P1) methods can attain the designed 2nd order and 3rd order accuracy in space for the velocity respectively. Moreover, the 3rd order rDG(P1P2)+CG(P1) method significantly outperforms its 2nd order rDG(P0P1)+CG(P1) and rDG(P1P1)+CG(P1) counterparts: being able to not only increase the accuracy of the velocity by one order but also improve the accuracy of the pressure.