A mass conserving mixed stress formulation for the Stokes equations

Jay Gopalakrishnan, Philip L. Lederer*, Joachim Schöberl

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

14 Citations (Scopus)


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 $H(\operatorname{curl} \operatorname{div})$. New matrix-valued finite elements having continuous ‘normal-tangential’ components are constructed to approximate functions in $H(\operatorname{curl} \operatorname{div})$. An error analysis concludes with optimal rates of convergence for errors in $u$ (measured in a discrete $H^1$-norm), errors in $\sigma $ (measured in $L^2$) and the pressure $p$ (also measured in $L^2$). The exact mass conservation property is directly related to another structure-preservation property called pressure robustness, as shown by pressure-independent velocity error estimates. The computational cost measured in terms of interface degrees of freedom is comparable to old and new Stokes discretizations.
Original languageEnglish
Pages (from-to)1838-1874
Number of pages37
JournalIMA Journal of Numerical Analysis
Issue number3
Publication statusPublished - 17 Jul 2020
Externally publishedYes


  • Incompressible flows
  • Mixed finite element methods
  • Stokes equations
  • n/a OA procedure


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