Divergence-free tangential finite element methods for incompressible flows on surfaces

Philip L. Lederer, Christoph Lehrenfeld*, Joachim Schöberl

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

20 Citations (Scopus)
29 Downloads (Pure)


In this work we consider the numerical solution of incompressible flows on two-dimensional manifolds. Whereas the compatibility demands of the velocity and the pressure spaces are known from the flat case one further has to deal with the approximation of a velocity field that lies only in the tangential space of the given geometry. Abandoning H1-conformity allows us to construct finite elements which are—due to an application of the Piola transformation—exactly tangential. To reintroduce continuity (in a weak sense) we make use of (hybrid) discontinuous Galerkin techniques. To further improve this approach, (Formula presented.) -conforming finite elements can be used to obtain exactly divergence-free velocity solutions. We present several new finite element discretizations. On a number of numerical examples we examine and compare their qualitative properties and accuracy.

Original languageEnglish
Pages (from-to)2503-2533
Number of pages31
JournalInternational journal for numerical methods in engineering
Issue number11
Publication statusPublished - 15 Jun 2020
Externally publishedYes


  • Divergence-conforming finite elements
  • incompressible Navier-Stokes equations
  • Piola transformation
  • Surface PDEs
  • Tangential vector field


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