427 research outputs found
On the role of the Helmholtz-decomposition in mixed methods for incompressible flows and a new variational crime
According to the Helmholtz decomposition, the irrotational parts of the momentum balance equations of the incompressible Navier-Stokes equations are balanced by the pressure gradient. Unfortunately, nearly all mixed methods for incompressible flows violate this fundamental property, resulting in the well-known numerical instability of poor mass conservation. The origin of this problem is the lack of L2-orthogonality between discretely divergence-free velocities and irrotational vector fields. In order to cure this, a new variational crime using divergence-free velocity reconstructions is proposed. Applying lowest order Raviart-Thomas velocity reconstructions to the nonconforming Crouzeix-Raviart element allows to construct a cheap flow discretization for general 2d and 3d simplex meshes that possesses the same advantageous robustness properties like divergence-free flow solvers. In the Stokes case, optimal a-priori error estimates for the velocity gradients and the pressure are derived. Moreover, the discrete velocity is independent of the continuous pressure. Several detailed linear and nonlinear numerical examples illustrate the theoretical findings
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Collision in a cross-shaped domain
In the numerical simulation of the incompressible Navier-Stokes
equations different numerical instabilities can occur. While instability in
the discrete velocity due to dominant convection and instability in the
discrete pressure due to a vanishing discrete LBB constant are well-known,
instability in the discrete velocity due to a poor mass conservation at high
Reynolds numbers sometimes seems to be underestimated. At least, when using
conforming Galerkin mixed finite element methods like the Taylor-Hood
element, the classical grad-div stabilization for enhancing discrete mass
conservation is often neglected in practical computations. Though simple
academic flow problems showing the importance of mass conservation are
well-known, these examples differ from practically relevant ones, since
specially designed force vectors are prescribed. Therefore we present a
simple steady Navier-Stokes problem in two space dimensions at Reynolds
number 1024, a colliding flow in a cross-shaped domain, where the instability
of poor mass conservation is studied in detail and where no force vector is
prescribed
Collision in a cross-shaped domain --- A steady 2D Navier--Stokes example demonstrating the importance of mass conservation in CFD
In the numerical simulation of the incompressible Navier-Stokes equations different numerical instabilities can occur. While instability in the discrete velocity due to dominant convection and instability in the discrete pressure due to a vanishing discrete LBB constant are well-known, instability in the discrete velocity due to a poor mass conservation at high Reynolds numbers sometimes seems to be underestimated. At least, when using conforming Galerkin mixed finite element methods like the Taylor-Hood element, the classical grad-div stabilization for enhancing discrete mass conservation is often neglected in practical computations. Though simple academic flow problems showing the importance of mass conservation are well-known, these examples differ from practically relevant ones, since specially designed force vectors are prescribed. Therefore we present a simple steady Navier-Stokes problem in two space dimensions at Reynolds number 1024, a colliding flow in a cross-shaped domain, where the instability of poor mass conservation is studied in detail and where no force vector is prescribed
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
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Guaranteed energy error estimators for a modified robust Crouzeix-Raviart Stokes element
This paper provides guaranteed upper energy error bounds for a modified
lowest-order nonconforming Crouzeix-Raviart finite element method for the
Stokes equations. The modification from [A. Linke 2014, On the role of the
Helmholtz-decomposition in mixed methods for incompressible flows and a new
variational crime] is based on the observation that only the divergence-free
part of the right-hand side should balance the vector Laplacian. The new
method has optimal energy error estimates and can lead to errors that are
smaller by several magnitudes, since the estimates are pressure-independent.
An efficient a posteriori velocity error estimator for the modified method
also should involve only the divergence-free part of the right-hand side.
Some designs to approximate the Helmholtz projector are compared and verified
by numerical benchmark examples. They show that guaranteed error control for
the modified method is possible and almost as sharp as for the unmodified
method
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Stabilized finite element schems for incompressible flow using Scott-Vogelius elements
We propose a stabilized finite element method based on the
Scott-Vogelius element in combination with either a local projection
stabilization or an edge oriented stabilization based on a penalization of
the gradient jumps over element edges. We prove a discrete inf-sup condition
leading to optimal a priori error estimates. The theoretical considerations
are illustrated by some numerical examples
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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
Uniform second order convergence of a complete flux scheme on unstructured 1D grids for a singularly perturbed advection-diffusion equation and some multidimensional extensions
The accurate and efficient discretization of singularly perturbed advection-diffusion equations on arbitrary 2D and 3D domains remains an open problem. An interesting approach to tackle this problem is the complete flux scheme (CFS) proposed by G. D. Thiart and further investigated by J. ten Thije Boonkkamp. For the CFS, uniform second order convergence has been proven on structured grids. We extend a version of the CFS to unstructured grids for a steady singularly perturbed advection-diffusion equation. By construction, the novel finite volume scheme is nodally exact in 1D for piecewise constant source terms. This property allows to use elegant continuous arguments in order to prove uniform second order convergence on unstructured one-dimensional grids. Numerical results verify the predicted bounds and suggest that by aligning the finite volume grid along the velocity field uniform second order convergence can be obtained in higher space dimensions as well
Guaranteed energy error estimators for a modified robust Crouzeix--Raviart Stokes element
This paper provides guaranteed upper energy error bounds for a modified lowest-order nonconforming Crouzeix--Raviart finite element method for the Stokes equations. The modification from [A. Linke 2014, On the role of the Helmholtz-decomposition in mixed methods for incompressible flows and a new variational crime] is based on the observation that only the divergence-free part of the right-hand side should balance the vector Laplacian. The new method has optimal energy error estimates and can lead to errors that are smaller by several magnitudes, since the estimates are pressure-independent. An efficient a posteriori velocity error estimator for the modified method also should involve only the divergence-free part of the right-hand side. Some designs to approximate the Helmholtz projector are compared and verified by numerical benchmark examples. They show that guaranteed error control for the modified method is possible and almost as sharp as for the unmodified method
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