908 research outputs found
Convergence analysis of the FEM approximation of the first order projection method for incompressible flows with and without the inf-sup condition
In this paper we obtain convergence results for the fully discrete projection method for the numerical approximation of the incompressible Navier–Stokes equations using a finite element approximation for the space discretization. We consider two situations. In the first one, the analysis relies on the satisfaction of the inf-sup condition for the velocity-pressure finite element spaces. After that, we study a fully discrete fractional step method using a Poisson equation for the pressure. In this case the velocity-pressure interpolations do not need to accomplish the inf-sup condition and in fact we consider the case in which equal velocity-pressure interpolation is used. Optimal convergence results in time and space have been obtained in both cases
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
Fully Discrete Approximations to the Time-Dependent Navier–Stokes Equations with a Projection Method in Time and Grad-Div Stabilization
This paper studies fully discrete approximations to the evolutionary Navier{
Stokes equations by means of inf-sup stable H1-conforming mixed nite elements
with a grad-div type stabilization and the Euler incremental projection method in
time. We get error bounds where the constants do not depend on negative powers
of the viscosity. We get the optimal rate of convergence in time of the projection
method. For the spatial error we get a bound O(hk) for the L2 error of the velocity,
k being the degree of the polynomials in the velocity approximation. We prove
numerically that this bound is sharp for this method.MINECO grant MTM2016-78995-P (AEI)Junta de Castilla y León grant VA024P17Junta de Castilla y León grant VA105G18MINECO grant MTM2015-65608-
Convergence to suitable weak solutions for a finite element approximation of the Navier-Stokes equations with numerical subgrid scale modeling
In this work we prove that weak solutions constructed by a variational
multiscale method are suitable in the sense of Scheffer. In order to prove this
result, we consider a subgrid model that enforces orthogonality between subgrid
and finite element components. Further, the subgrid component must be tracked
in time. Since this type of schemes introduce pressure stabilization, we have
proved the result for equal-order velocity and pressure finite element spaces
that do not satisfy a discrete inf-sup condition.Comment: 23 pages, no figure
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
Continuous, Semi-discrete, and Fully Discretized Navier-Stokes Equations
The Navier--Stokes equations are commonly used to model and to simulate flow
phenomena. We introduce the basic equations and discuss the standard methods
for the spatial and temporal discretization. We analyse the semi-discrete
equations -- a semi-explicit nonlinear DAE -- in terms of the strangeness index
and quantify the numerical difficulties in the fully discrete schemes, that are
induced by the strangeness of the system. By analyzing the Kronecker index of
the difference-algebraic equations, that represent commonly and successfully
used time stepping schemes for the Navier--Stokes equations, we show that those
time-integration schemes factually remove the strangeness. The theoretical
considerations are backed and illustrated by numerical examples.Comment: 28 pages, 2 figure, code available under DOI: 10.5281/zenodo.998909,
https://doi.org/10.5281/zenodo.99890
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