The analogue of grad–div stabilization in DG methods for incompressible flows: Limiting behavior and extension to tensor-product meshes

Abstract

Grad-div stabilization is a classical remedy in conforming mixed finite element methods for incompressible flow problems, for mitigating velocity errors that are sometimes called poor mass conservation. Such errors arise due to the relaxation of the divergence constraint in classical mixed methods, and are excited whenever the spatial discretization has to deal with comparably large and complicated pressures. In this contribution, an analogue of grad-div stabilization for Discontinuous Galerkin methods is studied. Here, the key is the penalization of the jumps of the normal velocities over facets of the triangulation, which controls the measure-valued part of the distributional divergence of the discrete velocity solution. Our contribution is twofold: first, we characterize the limit for arbitrarily large penalization parameters, which shows that the stabilized nonconforming Discontinuous Galerkin methods remain robust and accurate in this limit; second, we extend these ideas to the case of non-simplicial meshes; here, broken grad-div stabilization must be used in addition to the normal velocity jump penalization, in order to get the desired pressure robustness effect. The analysis is performed for the Stokes equations, and more complex flows and Crouzeix-Raviart elements are considered in numerical examples that also show the relevance of the theory in practical settings.

On high-order pressure-robust space discretisations, their advantages for incompressible high Reynolds number generalised Beltrami flows and beyond

“…Thus, pressure-robustness appears to be a prerequisite for accurate incompressible flow solvers at high Reynolds numbers. The arguments are supported by numerical analysis and numerical experiments.Indeed, the lack of pressure-robustness has been a rather hot research topic in the beginning of the history of finite element methods for CFD [55,31,62,38,25,35] -sometimes called poor mass conservation -and continued to be investigated for many years [33,57,34,60], often in connection with the so-called grad-div stabilisation [32,54,17,40,3,22]. Also, in the geophysical fluid dynamics community and in numerical astrophysics well-balanced schemes have been proposed to overcome similar issues for related Euler and shallow-water equations, especially in connection to nearly-hydrostatic and nearly-geostrophic flows; cf., for example, [20,21,11,44].However, only recently it was understood better that exactly the relaxation of the divergence constraint for incompressible flows, which was invented in classical mixed methods in order to construct discretely inf-sup stable discretisation schemes, introduces the lack of pressure-robustness, since it leads to a poor discretisation of the Helmholtz-Hodge projector [49].…”

Some continuous and discontinuous Galerkin methods and structure preservation for incompressible flows

“…2,26,27 Finally, for discontinuous Galerkin (dG) methods, a common technique for pressure robustness is penalizing the jump of the velocity normal component. 20,[28][29][30] An interesting result is reported in Reference 31, where an element-wise grad-div penalization was used on tensor product meshes for a nonisothermal flow, and an improvement of the mass conservation is observed for an inf-sup stable element pair of equal order. A discrete inf-sup condition involving the pressure jump is constructed in Reference 5 for both the steady incompressible Stokes and Navier-Stokes equations.…”

Assessment of high-order IMEX methods for incompressible flow

“…The explicit treatment of convection terms leads to a Courant-Friedrichs-Lewy (CFL) condition, which depends on the type and the order of the spatial discretization. For methods based on polynomial elements, the admissible time step scales approximately as ∆t ∼ ∆x /vP 2 where ∆x is the element length, v the fluid velocity and P the polynomial degree, see e.g. [28].…”

“…v ← vK 33: end procedure robustness a divergence/mass-flux stabilization is added as proposed in [2,25]. A detailed description of the DG-SEM is given in [53].…”