In this article, we consider exactly divergence-free H(div)-conforming finite element methods for time-dependent incompressible viscous flow problems. This is an extension of previous research concerning divergence-free H 1 -conforming methods. For the linearised Oseen case, the first semi-discrete numerical analysis for time-dependent flows is presented whereby special emphasis is put on pressure-and Reynolds-semi-robustness. For convection-dominated problems, the proposed method relies on a velocity jump upwind stabilisation which is not gradient-based. Complementing the theoretical results, H(div)-FEM are applied to the simulation of full nonlinear Navier-Stokes problems. Focussing on dynamic high Reynolds number examples with vortical structures, the proposed method proves to be capable of reliably handling the planar lattice flow problem, Kelvin-Helmholtz instabilities and freely decaying two-dimensional turbulence.

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 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, pressurerobust 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.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]. The reason is that the relaxation of the divergence constraint implies a relaxation of the L 2 -orthogonality between discretely divergence-free velocity test functions and arbitrary gradient fields. PRESSURE-ROBUSTNESS, HIGH REYNOLDS NUMBERS, BELTRAMI FLOWSWe only briefly remark that the question of an appropriate discretisation of the nonlinear convection term is intimately connected to the issue of numerical convection stabilisation techniques like upwinding or SUPG [56,14]. With the help of generalised Beltrami flows, we will demonstrate that in real-world flows the nonlinear convection term can be strong, even if the dynamics of the flow is not convectiondominated ...

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, ...

Summary The accurate numerical simulation of turbulent incompressible flows is a challenging topic in computational fluid dynamics. For discretisation methods to be robust in the underresolved regime, mass conservation and energy stability are key ingredients to obtain robust and accurate discretisations. Recently, two approaches have been proposed in the context of high‐order discontinuous Galerkin (DG) discretisations that address these aspects differently. On the one hand, standard L2‐based DG discretisations enforce mass conservation and energy stability weakly by the use of additional stabilisation terms. On the other hand, pointwise divergence‐free H(div)‐conforming approaches ensure exact mass conservation and energy stability by the use of tailored finite element function spaces. This work raises the question whether and to which extent these two approaches are equivalent when applied to underresolved turbulent flows. This comparative study highlights similarities and differences of these two approaches. The numerical results emphasise that both discretisation strategies are promising for underresolved simulations of turbulent flows due to their inherent dissipation mechanisms.

Two-dimensional Kelvin-Helmholtz instability problems are popular examples for assessing discretizations for incompressible flows at high Reynolds number. Unfortunately, the results in the literature differ considerably. This paper presents computational studies of a Kelvin-Helmholtz instability problem with high order divergence-free finite element methods. Reference results in several quantities of interest are obtained for three different Reynolds numbers up to the beginning of the final vortex pairing. A meshindependent prediction of the final pairing is not achieved due to the sensitivity of the considered problem with respect to small perturbations. A theoretical explanation of this sensitivity to small perturbations is provided based on the theory of self-organization of 2D turbulence. Possible sources of perturbations that arise in almost any numerical simulation are discussed.

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.

This article focusses on the analysis of a conforming finite element method for the time-dependent incompressible Navier-Stokes equations. For divergence-free approximations, in a semi-discrete formulation, we prove error estimates for the velocity that hold independently of both pressure and Reynolds number. Here, a key aspect is the use of the discrete Stokes projection for the error splitting. Optionally, edge-stabilisation can be included in the case of dominant convection. Emphasising the importance of conservation properties, the theoretical results are complemented with numerical simulations of vortex dynamics and laminar boundary layer flows.Mathematics Subject Classification: 65M12 · 65M15 · 65M60 · 76D05 · 76D10 · 76D17

This paper presents heavily grad-div and pressure jump stabilised, equal-and mixed-order discontinuous Galerkin finite element methods for non-isothermal incompressible flows based on the Oberbeck-Boussinesq approximation. In this framework, the enthalpy-porosity model for multiphase flow in melting and solidification problems can be employed. By considering the differentially heated cavity and the melting of pure gallium in a rectangular enclosure, it is shown that both boundary layers and sharp moving interior layers can be handled naturally by the proposed class of non-conforming methods. Due to the stabilising effect of the grad-div term and the robustness of discontinuous Galerkin methods, it is possible to solve the underlying problems accurately on coarse, non-adapted meshes. The interaction of heavy grad-div stabilisation and discontinuous Galerkin methods significantly improves the mass conservation properties and the overall accuracy of the numerical scheme which is observed for the first time. Hence, it is inferred that stabilised discontinuous Galerkin methods are highly robust as well as computationally efficient numerical methods to deal with natural convection problems arising in incompressible computational thermo-fluid dynamics.