# 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}
}
• Published 19 June 2018
• Mathematics
• IMA Journal of Numerical Analysis
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 $H(\operatorname{curl… 19 Citations ## Figures and Tables from this paper • Mathematics SIAM J. Numer. Anal. • 2020 A new discretization of a mixed formulation of the incompressible Stokes equations that includes symmetric viscous stresses is introduced that directly approximates the viscous fluid stress$\sigma$, enforcing its symmetry weakly. • Mathematics ArXiv • 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. • Shukai Du • Materials Science Journal 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. • 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 • Computer Science SIAM 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. • Computer Science J. 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. • Computer Science International 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. ## References SHOWING 1-10 OF 44 REFERENCES • Mathematics, Computer Science • 2016 A discontinuous Galerkin method for the discretization of the Stokes problem is considered and it is shown that the considered method is uniformly stable with respect to the polynomial order$k$and provides optimal error estimates. • Computer Science SIAM Rev. • 2017 Several approaches for improving the discrete mass balance or even for computing divergence-free solutions will be discussed: grad-div stabilization, higher order mixed methods derived on the basis of an exact de Rham complex,$H(div)\$-conforming finite ...
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