Robust arbitrary order mixed finite element methods for the incompressible Stokes equations

@inproceedings{Linke2014RobustAO,
  title={Robust arbitrary order mixed finite element methods for the incompressible Stokes equations},
  author={Alexander Linke and Gunar Matthies and Lutz Tobiska},
  year={2014}
}
Standard mixed finite element methods for the incompressible Navier–Stokes equations that relax the divergence constraint are not robust against large irrotational forces in the momentum balance and the velocity error depends on the continuous pressure. This robustness issue can be completely cured by using divergence-free mixed finite elements which deliver pressure-independent velocity error estimates. However, the construction of H-conforming, divergence-free mixed finite element methods is… 

Tables from this paper

On the Divergence Constraint in Mixed Finite Element Methods for Incompressible Flows
TLDR
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 ...
Divergence-free Reconstruction Operators for Pressure-Robust Stokes Discretizations with Continuous Pressure Finite Elements
TLDR
This contribution extends the idea of modification only in the right-hand side of a Stokes discretization to low and high order Taylor--Hood and mini elements, which have continuous discrete pressures.
Towards Pressure-Robust Mixed Methods for the Incompressible Navier–Stokes Equations
TLDR
This contribution reviews classical mixed methods for the incompressible Navier–Stokes equations that relax the divergence constraint and are discretely inf-sup stable and pressure-robust, simultaneously, to allow in future to reduce the approximation order of the discretizations used in computational practice, without compromising the accuracy.
On Really Locking-Free Mixed Finite Element Methods for the Transient Incompressible Stokes Equations
TLDR
Semidiscrete and fully discrete a priori velocity and pressure error estimates are derived, which show remarkable robustness properties.
A high-order discontinuous Galerkin pressure robust splitting scheme for incompressible flows
TLDR
A novel postprocessing technique in the projection step of the splitting scheme that reconstructs the Helmholtz flux in H(\text{div}) is presented, and it is demonstrated that a robust DG method for underresolved turbulent incompressible flows can be realized.
Pressure-robustness in quasi-optimal a priori estimates for the Stokes problem
TLDR
Pressure-robustness of the estimate is achieved by the fact that the new estimate only depends on the Helmholtz--Hodge projector of the data term and not on the $L^2$ norm of the entire data term.
A nonconforming pressure-robust finite element method for the Stokes equations on anisotropic meshes
TLDR
The novelty of the present contribution is that the reconstruction approach for the Crouzeix-Raviart method, which has a stable Fortin operator on arbitrary meshes, is combined with results on the interpolation error on anisotropic elements for reconstruction operators of Raviart-Thomas and Brezzi-Douglas-Marini type, generalizing the method to a large class of anisotrop triangulations.
Quasi-optimal and pressure-robust discretizations of the Stokes equations by new augmented Lagrangian formulations
TLDR
A discretization that is quasi-optimal and pressure robust, in the sense that the velocity H^1-error is proportional to the best $H^1$-error to the analytical velocity, shows that such a property can be achieved without using conforming and divergence-free pairs.
...
...

References

SHOWING 1-10 OF 50 REFERENCES
ISOGEOMETRIC DIVERGENCE-CONFORMING B-SPLINES FOR THE STEADY NAVIER–STOKES EQUATIONS
TLDR
A comprehensive suite of numerical experiments are presented which indicate optimal convergence rates for both the discrete velocity and pressure fields for general configurations, suggesting that the a priori estimates may be conservative.
Aspects of Finite Element Discretizations for Solving the Boussinesq Approximation of the Navier-Stokes Equations
We consider stable discretizations for solving the Boussinesq approximation of the stationary, incompressible Navier-Stokes equations in the twodimensional case. For the continuous problem the right
Two classes of mixed finite element methods
Optimal and Pressure-Independent $$L^2$$ Velocity Error Estimates for a Modified Crouzeix-Raviart Element with BDM Reconstructions
TLDR
A more sophisticated variational crime employing the lowest-order BDM element is proposed, which allows proving an optimal pressure-independent \(L^2\) velocity error and numerical examples confirm the analytical results.
Robust Numerical Methods for Singularly Perturbed Differential Equations: Convection-Diffusion-Reaction and Flow Problems
Ordinary Differential Equations.- The Analytical Behaviour of Solutions.- Numerical Methods for Second-Order Boundary Value Problems.- Parabolic Initial-Boundary Value Problems in One Space
A Note on Discontinuous Galerkin Divergence-free Solutions of the Navier–Stokes Equations
TLDR
A class of discontinuous Galerkin methods for the incompressible Navier–Stokes equations yielding exactly divergence-free solutions is presented, which are locally conservative, energy-stable, and optimally convergent.
Conforming and divergence-free Stokes elements on general triangular meshes
TLDR
A family of conforming elements for the Stokes problem on general triangular meshes in two dimensions is presented and it is shown how the proposed elements are related to a class of C 1 elements through the use of a discrete de Rham complex.
A new family of stable mixed finite elements for the 3D Stokes equations
TLDR
This paper is devoted to proving that the mixed finite elements of this P k -P k-1 type when k > 3 satisfy the stability condition-the Babuska-Brezzi inequality on macro-tetrahedra meshes where each big tetrahedron is subdivided into four subtetrahedral meshes.
...
...