# A mass conserving mixed stress formulation for the Stokes equations

@article{Gopalakrishnan2018AMC, title={A mass conserving mixed stress formulation for the Stokes equations}, author={Jay Gopalakrishnan and Philip L. Lederer and Joachim Schoberl}, journal={IMA Journal of Numerical Analysis}, year={2018} }

We propose stress formulation of the Stokes equations. The velocity $u$ is approximated with $H(\operatorname{div})$-conforming finite elements providing exact mass conservation. While many standard methods use $H^1$-conforming spaces for the discrete velocity $H(\operatorname{div})$-conformity fits the considered variational formulation in this work. A new stress-like variable $\sigma $ equalling the gradient of the velocity is set within a new function space $H(\operatorname{curl…

## 19 Citations

### A Mass Conserving Mixed Stress Formulation for Stokes Flow with Weakly Imposed Stress Symmetry

- MathematicsSIAM J. Numer. Anal.
- 2020

A new discretization of a mixed formulation of the incompressible Stokes equations that includes symmetric viscous stresses is introduced that directly approximates the viscous fluid stress $\sigma$, enforcing its symmetry weakly.

### Uniformly well-posed hybridized discontinuous Galerkin/hybrid mixed discretizations for Biot's consolidation model

- Computer ScienceArXiv
- 2020

### Minimal order H(div)-conforming velocity-vorticity approximations for incompressible fluids

- MathematicsArXiv
- 2021

A novel minimal order hybrid Discontinuous Galerkin (HDG) and a novel mass conserving mixed stress (MCS) method for the approximation of incompressible flows and a new Korn-like inequality for vectorvalued element-wise H1 and normal continuous functions for the stability analysis are introduced.

### HDG Methods for Stokes Equation Based on Strong Symmetric Stress Formulations

- Materials ScienceJournal of Scientific Computing
- 2020

This work proposes a hybridizable discontinuous Galerkin (HDG) method for Stokes equation based on strong symmetric stress formulations and proves that both methods are optimal for all variables and achieve super-convergence for the numerical trace.

### Divergence-conforming velocity and vorticity approximations for incompressible fluids obtained with minimal inter-element coupling

- Mathematics
- 2021

. We introduce two new lowest order methods, a mixed method, and a hybrid Discontinuous Galerkin (HDG) method, for the approximation of incompressible flows. Both methods use divergence-conforming…

### A pressure-robust embedded discontinuous Galerkin method for the Stokes problem by reconstruction operators

- Computer ScienceSIAM J. Numer. Anal.
- 2020

A local reconstruction operator that maps discretely divergence- free test functions to exactly divergence-free test functions and 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.

### Guaranteed upper bounds for the velocity error of pressure-robust Stokes discretisations

- Computer ScienceJ. Num. Math.
- 2022

The first main result of the paper is a framework with relaxed constraints on the primal and dual method that enables to use a recently developed mass conserving mixed stress discretisation for the design of equilibrated fluxes and to obtain pressure-independent guaranteed upper bounds for any pressure-robust primal discretisations.

### Divergence‐free tangential finite element methods for incompressible flows on surfaces

- Computer ScienceInternational journal for numerical methods in engineering
- 2020

This work considers the numerical solution of incompressible flows on two‐dimensional manifolds and presents several new finite element discretizations, including H(divΓ) ‐conforming finite elements can be used to obtain exactly divergence‐free velocity solutions.

### On pressure robustness and independent determination of displacement and pressure in incompressible linear elasticity

- MathematicsComputer Methods in Applied Mechanics and Engineering
- 2023

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