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arXiv · 1909.06229

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

Philip L. Lederer·Christoph Lehrenfeld·Joachim Schöberl
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Abstract

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 $H^1$-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, $H(\operatorname{div}_{\Gamma})$-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.

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Philip L. Lederer, Christoph Lehrenfeld, Joachim Schöberl. 2019-09-13. Divergence-free tangential finite element methods for incompressible flows on surfaces. https://doi.org/10.1002/nme.6317

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