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

An improved understanding of the divergence-free constraint for the incompressible Navier--Stokes equations leads to the observation that a semi-norm and corresponding equivalence classes of forces are fundamental for their nonlinear dynamics. The recent concept of {\em pressure-robustness} allows to distinguish between space discretisations that discretise these equivalence classes appropriately or not. This contribution compares the accuracy of pressure-robust and non-pressure-robust space discretisations for transient high Reynolds number flows, starting from the observation that in generalised Beltrami flows the nonlinear convection term is balanced by a strong pressure gradient. Then, pressure-robust methods are shown to outperform comparable non-pressure-robust space discretisations. Indeed, pressure-robust methods of formal order $k$ are comparably accurate than non-pressure-robust methods of formal order $2k$ on coarse meshes. Investigating the material derivative of incompressible Euler flows, it is conjectured that strong pressure gradients are typical for non-trivial high Reynolds number flows. Connections to vortex-dominated flows are established. 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.Comment: 43 pages, 18 figures, 2 table

Towards pressure-robust mixed methods for the incompressible Navier--Stokes equations

In this contribution, classical mixed methods for the incompressible Navier-Stokes equations that relax the divergence constraint and are discretely inf-sup stable, are reviewed. Though the relaxation of the divergence constraint was claimed to be harmless since the beginning of the 1970ies, Poisson locking is just replaced by another more subtle kind of locking phenomenon, which is sometimes called poor mass conservation. Indeed, divergence-free mixed methods and classical mixed methods behave qualitatively in a different way: divergence-free mixed methods are pressure-robust, which means that, e.g., their velocity error is independent of the continuous pressure. The lack of pressure-robustness in classical mixed methods can be traced back to a consistency error of an appropriately defined discrete Helmholtz projector. Numerical analysis and numerical examples reveal that really locking-free mixed methods must be discretely inf-sup stable and pressure-robust, simultaneously. Further, a recent discovery shows that locking-free, pressure-robust mixed methods do not have to be divergence-free. Indeed, relaxing the divergence constraint in the velocity trial functions is harmless, if the relaxation of the divergence constraint in some velocity test functions is repaired, accordingly

On the divergence constraint in mixed finite element methods for incompressible flows

The divergence constraint of the incompressible Navier--Stokes equations is revisited in the mixed finite element framework. While many stable and convergent mixed elements have been developed throughout the past four decades, most classical methods relax the divergence constraint and only enforce the condition discretely. As a result, these methods introduce a pressure-dependent consistency error which can potentially pollute the computed velocity. These methods are not robust in the sense that a contribution from the right-hand side, which influences only the pressure in the continuous equations, impacts both velocity and pressure in the discrete equations. This paper reviews the theory and practical implications of relaxing the divergence constraint. Several approaches for improving the discrete mass balance or even for computing divergence-free solutions will be discussed: grad-div stabilization, higher order mixed methods derived on the basis of an exact de Rham complex, \bH(\mathrm{div})-conforming finite elements, and mixed methods with an appropriate reconstruction of the test functions. Numerical examples illustrate both the potential effects of using non-robust discretizations and the improvements obtained by utilizing pressure-robust discretizations

Optimal and pressure-independent L2 velocity error estimates for a modified Crouzeix-Raviart Stokes element with BDM reconstructions

Nearly all inf-sup stable mixed finite elements for the incompressible Stokes equations relax the divergence constraint. The price to pay is that a priori estimates for the velocity error become pressure-dependent, while divergence-free mixed finite elements deliver pressure-independent estimates. A recently introduced new variational crime using lowest-order Raviart-Thomas velocity reconstructions delivers a much more robust modified Crouzeix-Raviart element, obeying an optimal pressure-independent discrete H1 velocity estimate. Refining this approach, a more sophisticated variational crime employing the lowest-order BDM element is proposed, which also allows proving an optimal pressure-independent L2 velocity error. Numerical examples confirm the analysis and demonstrate the improved robustness in the Navier-Stokes case

Pressure-robustness and discrete Helmholtz projectors in mixed finite element methods for the incompressible Navier--Stokes equations

Recently, it was understood how to repair a certain L2-orthogonality of discretely-divergence-free vector fields and gradient fields such that the velocity error of inf-sup stable discretizations for the incompressible Stokes equations becomes pressure-independent. These new 'pressure-robust' Stokes discretizations deliver a small velocity error, whenever the continuous velocity field can be well approximated on a given grid. On the contrary, classical inf-sup stable Stokes discretizations can guarantee a small velocity error only, when both the velocity and the pressure field can be approximated well, simultaneously. In this contribution, 'pressure-robustness' is extended to the time-dependent Navier--Stokes equations. In particular, steady and time-dependent potential flows are shown to build an entire class of benchmarks, where pressure-robust discretizations can outperform classical approaches significantly. Speedups will be explained by a new theoretical concept, the 'discrete Helmholtz projector' of an inf-sup stable discretization. Moreover, different discrete nonlinear convection terms are discussed, and skew-symmetric pressure-robust discretizations are proposed

On really locking-free mixed finite element methods for the transient incompressible Stokes equations

Inf-sup stable mixed methods for the steady incompressible Stokes equations that relax the divergence constraint are often claimed to deliver locking-free discretizations. However, this relaxation leads to a pressure-dependent contribution in the velocity error, which is proportional to the inverse of the viscosity, thus giving rise to a (different) locking phenomenon. However, a recently proposed modification of the right hand side alone leads to a discretization that is really locking-free, i.e., its velocity error converges with optimal order and is independent of the pressure and the smallness of the viscosity. In this contribution, we extend this approach to the transient incompressible Stokes equations, where besides the right hand side also the velocity time derivative requires an improved space discretization. Semi-discrete and fully-discrete a-priori velocity and pressure error estimates are derived, which show beautiful robustness properties. Two numerical examples illustrate the superior accuracy of pressure-robust space discretizations in the case of small viscosities