162 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
Pressure-robustness in the context of optimal control
This paper studies the benefits of pressure-robust discretizations in the scope of optimal control of incompressible flows. Gradient forces that may appear in the data can have a negative impact on the accuracy of state and control and can only be correctly balanced if their L2-orthogonality onto discretely divergence-free test functions is restored. Perfectly orthogonal divergence-free discretizations or divergence-free reconstructions of these test functions do the trick and lead to much better analytic a priori estimates that are also validated in numerical examples
An embedded--hybridized discontinuous Galerkin finite element method for the Stokes equations
We present and analyze a new embedded--hybridized discontinuous Galerkin
finite element method for the Stokes problem. The method has the attractive
properties of full hybridized methods, namely an -conforming
velocity field, pointwise satisfaction of the continuity equation and \emph{a
priori} error estimates for the velocity that are independent of the pressure.
The embedded--hybridized formulation has advantages over a full hybridized
formulation in that it has fewer global degrees-of-freedom for a given mesh and
the algebraic structure of the resulting linear system is better suited to fast
iterative solvers. The analysis results are supported by a range of numerical
examples that demonstrate rates of convergence, and which show computational
efficiency gains over a full hybridized formulation
Divergence-preserving reconstructions on polygons and a really pressure-robust virtual element method for the Stokes problem
Non divergence-free discretisations for the incompressible Stokes problem may suffer from a lack of pressure-robustness characterised by large discretisations errors due to irrotational forces in the momentum balance. This paper argues that also divergence-free virtual element methods (VEM) on polygonal meshes are not really pressure-robust as long as the right-hand side is not discretised in a careful manner. To be able to evaluate the right-hand side for the testfunctions, some explicit interpolation of the virtual testfunctions is needed that can be evaluated pointwise everywhere. The standard discretisation via an L2 -bestapproximation does not preserve the divergence and so destroys the orthogonality between divergence-free testfunctions and possibly eminent gradient forces in the right-hand side. To repair this orthogonality and restore pressure-robustness another divergence-preserving reconstruction is suggested based on Raviart--Thomas approximations on local subtriangulations of the polygons. All findings are proven theoretically and are demonstrated numerically in two dimensions. The construction is also interesting for hybrid high-order methods on polygonal or polyhedral meshes
Refined a posteriori error estimation for classical and pressure-robust Stokes finite element methods
Recent works showed that pressure-robust modifications of mixed finite
element methods for the Stokes equations outperform their standard versions in
many cases. This is achieved by divergence-free reconstruction operators and
results in pressure independent velocity error estimates which are robust with
respect to small viscosities. In this paper we develop a posteriori error
control which reflects this robustness.
The main difficulty lies in the volume contribution of the standard
residual-based approach that includes the -norm of the right-hand side.
However, the velocity is only steered by the divergence-free part of this
source term. An efficient error estimator must approximate this divergence-free
part in a proper manner, otherwise it can be dominated by the pressure error.
To overcome this difficulty a novel approach is suggested that uses arguments
from the stream function and vorticity formulation of the Navier--Stokes
equations. The novel error estimators only take the of the
right-hand side into account and so lead to provably reliable, efficient and
pressure-independent upper bounds in case of a pressure-robust method in
particular in pressure-dominant situations. This is also confirmed by some
numerical examples with the novel pressure-robust modifications of the
Taylor--Hood and mini finite element methods
A pressure-robust embedded discontinuous Galerkin method for the Stokes problem by reconstruction operators
The embedded discontinuous Galerkin (EDG) finite element method for the
Stokes problem results in a point-wise divergence-free approximate velocity on
cells. However, the approximate velocity is not H(div)-conforming and it can be
shown that this is the reason that the EDG method is not pressure-robust, i.e.,
the error in the velocity depends on the continuous pressure. In this paper we
present a local reconstruction operator that maps discretely divergence-free
test functions to exactly divergence-free test functions. This local
reconstruction operator restores pressure-robustness by only changing the right
hand side of the discretization, similar to the reconstruction operator
recently introduced for the Taylor--Hood and mini elements by Lederer et al.
(SIAM J. Numer. Anal., 55 (2017), pp. 1291--1314). We present an a priori error
analysis of the discretization showing optimal convergence rates and
pressure-robustness of the velocity error. These results are verified by
numerical examples. The motivation for this research is that the resulting EDG
method combines the versatility of discontinuous Galerkin methods with the
computational efficiency of continuous Galerkin methods and accuracy of
pressure-robust finite element methods
On pressure robustness and independent determination of displacement and pressure in incompressible linear elasticity
We investigate the possibility to determine the divergence-free displacement
\emph{independently} from the pressure reaction for a class of
boundary value problems in incompressible linear elasticity. If not possible,
we investigate if it is possible to determine it \emph{pressure robustly}, i.e.
pollution free from the pressure reaction.
For convex domains there is but one variational boundary value problem among
the investigated that allows the independent determination. It is the one with
essential no-penetration conditions combined with homogeneous tangential
traction conditions.
Further, in most but not all investigated cases, the weakly divergence-free
displacement can be computed pressure robustly provided the total body force is
decomposed into its direct sum of divergence- and rotation-free components
using a Helmholtz decomposition. The elasticity problem is solved using these
components as separate right-hand sides. The total solution is obtained using
the superposition principle.
We employ a higher-order finite element formulation with
discontinuous pressure elements. It is \emph{inf-sup} stable for polynomial
degree but not pressure robust by itself.
We propose a three step procedure to solve the elasticity problem preceded by
the Helmholtz decomposition of the total body force. The extra cost for the
three-step procedure is essentially the cost for the Helmholtz decomposition of
the assembled total body force, and the small cost of solving the elasticity
problem with one extra right-hand side. The results are corroborated by
theoretical derivations as well as numerical results.Comment: 31 pages, 31 references and 48 figure
Analysis of an exactly mass conserving space-time hybridized discontinuous Galerkin method for the time-dependent Navier--Stokes equations
We introduce and analyze a space-time hybridized discontinuous Galerkin
method for the evolutionary Navier--Stokes equations. Key features of the
numerical scheme include point-wise mass conservation, energy stability, and
pressure robustness. We prove that there exists a solution to the resulting
nonlinear algebraic system in two and three spatial dimensions, and that this
solution is unique in two spatial dimensions under a small data assumption. A
priori error estimates are derived for the velocity in a mesh-dependent energy
norm
- …