5 research outputs found
Guaranteed error control for the pseudostress approximation of the Stokes equations
The pseudostress approximation of the Stokes equations rewrites the stationary Stokes equations with pure (but possibly inhomogeneous) Dirichlet boundary conditions as another (equivalent) mixed scheme based on a stress in H(div) and the velocity in . Any standard mixed finite element function space can be utilized for this mixed formulation, e.g. the Raviart-Thomas discretization which is related to the Crouzeix-Raviart nonconforming finite element scheme in the lowest-order case. The effective and guaranteed a posteriori error control for this nonconforming velocity-oriented discretization can be generalized to the error control of some piecewise quadratic velocity approximation that is related to the discrete pseudostress. The analysis allows for local inf-sup constants which can be chosen in a global partition to improve the estimation. Numerical examples provide strong evidence for an effective and guaranteed error control with very small overestimation factors even for domains with large anisotropy
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
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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
L2-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 NavierStokes equations. The novel error estimators only
take the curl of the righthand 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 TaylorHood and mini finite element methods
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On the divergence constraint in mixed finite element methods for incompressible flows
The divergence constraint of the incompressible Navier-Stokes equations
is revisited in the mixed finite element framework. While many stable and
convergent mixed elements have been developed throughout the past four
decades, most classical methods relax the divergence constraint and only
enforce the condition discretely. As a result, these methods introduce a
pressure-dependent consistency error which can potentially pollute the
computed velocity. These methods are not robust in the sense that a
contribution from the right-hand side, which in fluences only the pressure in
the continuous equations, impacts both velocity and pressure in the discrete
equations. This paper reviews the theory and practical implications of
relaxing the divergence constraint. Several approaches for improving the
discrete mass balance or even for computing divergence-free solutions will be
discussed: grad-div stabilization, higher order mixed methods derived on the
basis of an exact de Rham complex, H(div)-conforming finite elements, and
mixed methods with an appropriate reconstruction of the test functions.
Numerical examples illustrate both the potential effects of using non-robust
discretizations and the improvements obtained by utilizing pressure-robust
discretizations
On the divergence constraint in mixed finite element methods for incompressible flows
The divergence constraint of the incompressible Navier--Stokes equations
is revisited in the mixed finite element framework.
While many stable and convergent mixed elements have been developed throughout the past
four decades,
most classical methods relax the divergence constraint and
only enforce
the condition discretely. As a result, these methods
introduce a pressure-dependent consistency error which can potentially
pollute the computed velocity. These methods are not robust in the
sense that a contribution from the right-hand side, which influences only the pressure
in the continuous equations, impacts both velocity and pressure
in the discrete equations.
This paper reviews the
theory and practical implications of relaxing
the divergence constraint. Several approaches for improving the
discrete mass balance or even for computing divergence-free solutions will be
discussed: grad-div stabilization, higher order mixed methods derived on the basis
of an exact de Rham complex, \bH(\mathrm{div})-conforming finite elements, and
mixed methods with an appropriate reconstruction of the test functions.
Numerical examples illustrate both the potential effects of using non-robust
discretizations and the improvements obtained by utilizing pressure-robust discretizations