On the dissipation of conforming and discontinuous Galerkin schemes for the incompressible Navier–Stokes equations

In this paper, we improve the numerical performance of the classical conforming finite element schemes for the time-dependent incompressible Navier–Stokes equations by adding dissipation. This is a physics-inspired approach, and the dissipative terms are constructed through the discontinuity of numerical quantities across interior edges and, therefore, decouple the space and time discretizations when compared with the streamline-upwind Petrov–Galerkin for the time-marching methods. In particular, the order of h (edge diameter) in the dissipative terms is motivated by the energy stability and error equation associated with the unsteady problem. Furthermore, we point out that the added dissipation may also be viewed as an alternative for the grad-div stabilization from the physical approach in the unsteady problem. The added dissipation is naturally within the framework of the variational multiscale and thus could serve as implicit subgrid-scale models in large eddy simulations. Numerical experiments with a jump of the gradient are performed. In addition, we test the ideas with the discontinuous Galerkin formulations. Numerical results indicate that our suggested dissipation is helpful in reducing numerical errors and is competitive when compared with other conventional stabilization available in the literature. Finally, we show that the changes in the physical role of the same terms may significantly change their corresponding numerical behaviors through examples on the steady problems.