Towards computable flows and robust estimates for inf-sup stable FEM applied to the time-dependent incompressible Navier–Stokes equations

Abstract

Inf-sup stable FEM applied to time-dependent incompressible Navier-Stokes flows are considered. The focus lies on robust estimates for the kinetic and dissipation energies in a twofold sense. Firstly, pressure-robustness ensures the fulfilment of a fundamental invariance principle and velocity error estimates are not corrupted by the pressure approximability. Secondly, Re-semi-robustness means that constants appearing on the right-hand side of kinetic and dissipation energy error estimates (including Gronwall constants) do not explicitly depend on the Reynolds number. Such estimates rely on the essential regularity assumption ∇u ∈ L 1 (0, T ; L ∞ (Ω )) which is discussed in detail. In the sense of best practice, we review and establish pressure-and Re-semi-robust estimates for pointwise divergence-free H 1 -conforming FEM (like Scott-Vogelius pairs or certain isogeometric based FEM) and pointwise divergence-free H(div)-conforming discontinuous Galerkin FEM. For convection-dominated problems, the latter naturally includes an upwind stabilisation for the velocity which is not gradient-based.Keywords time-dependent incompressible flow · Re-semi-robust error estimates · pressure-robustness · inf-sup stable methods · exactly divergence-free FEMA relatively new aspect in the FE analysis applied to incompressible flows is 'pressurerobustness' [41]. In its most general form, pressure-robustness of a numerical method is defined by its ability to fulfil the following requirement: if the exact solution u u u of (1) belongs to the approximation space V V V h , i.e. if u u u ∈ V V V h , then the discrete solution u u u h coincides with the exact one, that is, u u u h = u u u. In certain physical regimes of the incompressible Navier-Stokes equations -i.e., in certain benchmarks -pressure-robustness allows to use less expensive discretisation schemes without losing accuracy [49,1]. As a consequence, the following fundamental invariance principle transfers from the continuous level to the discretised case: Replacing the source term f f f by f f f + ∇ψ changes the solution (u u u, p) to (u u u, p + ψ). For example, in a potential flow, (u u u · · · ∇)u u u can be very large but it is a gradient and therefore balanced by the pressure gradient and thus does not have any impact on the velocity field. Only recently it has been shown that high Reynolds number potential flows are really challenging for the numerical solution with standard low-order Galerkin-FEM [50,41].A well-known important consequence for methods which are not pressure-robust is that already for the steady incompressible Stokes problem the velocity error estimates for kinetic and dissipation energies are corrupted by the pressure approximability multiplied by ν −1/2 [41,49]. Note that the mechanism responsible for the excitation of this kind of numerical error is a completely linear phenomenon. Exactly divergence-free FEM are naturally pressure-robust, but classical inf-sup stable velocity-pressure pairs like Taylor-Hood FEM are usually not pressure-robust. In fact, ...

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

“…Finite element error analysis for the L 2 -DG method can be found, for example, in [56,24]. For the H(div)-DG method, we refer to [60]. In the DG setting, discretely divergence-free functions are defined using the pressure-velocity coupling b h by…”

“…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].…”

On reference solutions and the sensitivity of the 2D Kelvin–Helmholtz instability problem

“…Remark 5 in [30]. We choose this method as it -compared to other standard discretization approachescombines features such as high order accuracy, important global and local conservation properties, energystability, polynomial, pressure and Re-semi-robustness, a minimal amount of numerical dissipation and computational efficiency [30,29,42]. Especially the combination of the robustness properties is hardly seen in other numerical discretization schemes.…”

Numerical benchmarking of fluid-rigid body interactions

“…The resulting spatial discretisation has several benefits such as energy stability and pressure robustness [25,26] while allowing for efficient and high order accurate implementations [24,27].…”