224 research outputs found
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 are comparably accurate than non-pressure-robust
methods of formal order 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 computable flows and robust estimates for inf-sup stable FEM applied to the time-dependent incompressible Navier--Stokes equations
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 an essential regularity assumption for the gradient of the velocity, 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 H1-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
Finite elements for scalar convection-dominated equations and incompressible flow problems - A never ending story?
The contents of this paper is twofold. First, important recent results concerning finite element
methods for convection-dominated problems and incompressible flow problems are described that
illustrate the activities in these topics. Second, a number of, in our opinion, important problems in
these fields are discussed
On the convergence order of the finite element error in the kinetic energy for high Reynolds number incompressible flows
The kinetic energy of a flow is proportional to the square of the norm of the velocity. Given a sufficient regular velocity field and a velocity finite element space with polynomials of degree , then the best approximation error in is of order . In this survey, the available finite element error analysis for the velocity error in is reviewed, where is a final time. Since in practice the case of small viscosity coefficients or dominant convection is of particular interest, which may result in turbulent flows, robust error estimates are considered, i.e., estimates where the constant in the error bound does not depend on inverse powers of the viscosity coefficient. Methods for which robust estimates can be derived enable stable flow simulations for small viscosity coefficients on comparatively coarse grids, which is often the situation encountered in practice. To introduce stabilization techniques for the convection-dominated regime and tools used in the error analysis, evolutionary linear convectionâdiffusion equations are studied at the beginning. The main part of this survey considers robust finite element methods for the incompressible NavierâStokes equations of order , , and for the velocity error in . All these methods are discussed in detail. In particular, a sketch of the proof for the error bound is given that explains the estimate of important terms which determine finally the order of convergence. Among them, there are methods for infâsup stable pairs of finite element spaces as well as for pressure-stabilized discretizations. Numerical studies support the analytic results for several of these methods. In addition, methods are surveyed that behave in a robust way but for which only a non-robust error analysis is available. The conclusion of this survey is that the problem of whether or not there is a robust method with optimal convergence order for the kinetic energy is still open
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
Longer time accuracy for incompressible Navier-Stokes simulations with the EMAC formulation
In this paper, we consider the recently introduced EMAC formulation for the
incompressible Navier-Stokes (NS) equations, which is the only known NS
formulation that conserves energy, momentum and angular momentum when the
divergence constraint is only weakly enforced. Since its introduction, the EMAC
formulation has been successfully used for a wide variety of fluid dynamics
problems. We prove that discretizations using the EMAC formulation are
potentially better than those built on the commonly used skew-symmetric
formulation, by deriving a better longer time error estimate for EMAC: while
the classical results for schemes using the skew-symmetric formulation have
Gronwall constants dependent on with the Reynolds
number, it turns out that the EMAC error estimate is free from this explicit
exponential dependence on the Reynolds number. Additionally, it is demonstrated
how EMAC admits smaller lower bounds on its velocity error, since {incorrect
treatment of linear momentum, angular momentum and energy induces} lower bounds
for velocity error, and EMAC treats these quantities more accurately.
Results of numerical tests for channel flow past a cylinder and 2D
Kelvin-Helmholtz instability are also given, both of which show that the
advantages of EMAC over the skew-symmetric formulation increase as the Reynolds
number gets larger and for longer simulation times.Comment: 21 pages, 5 figure
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
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