# 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…

## 72 Citations

### Divergence-free Reconstruction Operators for Pressure-Robust Stokes Discretizations with Continuous Pressure Finite Elements

- MathematicsSIAM J. Numer. Anal.
- 2017

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

- Computer ScienceComput. Methods Appl. Math.
- 2018

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.

### Pressure-robustness and discrete Helmholtz projectors in mixed finite element methods for the incompressible Navier--Stokes equations

- Computer Science
- 2016

### On Really Locking-Free Mixed Finite Element Methods for the Transient Incompressible Stokes Equations

- Computer ScienceSIAM J. Numer. Anal.
- 2018

Semidiscrete and fully discrete a priori velocity and pressure error estimates are derived, which show remarkable robustness properties.

### Analysis and computation of a pressure-robust method for the rotation form of the stationary incompressible Navier-Stokes equations by using high-order finite elements

- MathematicsComput. Math. Appl.
- 2022

### A high-order discontinuous Galerkin pressure robust splitting scheme for incompressible flows

- Computer ScienceArXiv
- 2019

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.

### Refined a posteriori error estimation for classical and pressure-robust Stokes finite element methods

- Computer ScienceNumerische Mathematik
- 2019

A posteriori error control is developed which reflects pressure-robust modifications of mixed finite element methods for the Stokes equations which are robust with respect to small viscosities and leads to provably reliable, efficient and pressure-independent upper bounds in case of a pressure-Robust method.

### Pressure-robustness in quasi-optimal a priori estimates for the Stokes problem

- MathematicsArXiv
- 2019

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

- Computer ScienceArXiv
- 2020

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

- Computer Science
- 2019

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.

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