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

@article{Lederer2020DivergencefreeTF, title={Divergence‐free tangential finite element methods for incompressible flows on surfaces}, author={Philip L. Lederer and Christoph Lehrenfeld and Joachim Sch{\"o}berl}, journal={International Journal for Numerical Methods in Engineering}, year={2020}, volume={121}, pages={2503 - 2533} }

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…

## 15 Citations

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This work first approximate the Killing fields via a Stokes eigenvalue problem and then gives a method which is asymptotically guaranteed to correctly exclude them from the solution, which exactly satisfies the incompressibility constraint in the surface Stokes problem.

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A general abstract framework applicable to such nonconforming discretizations of eigenproblems is presented and error bounds both for eigenvalue and eigenvector approximations are derived that depend on certain consistency and approximability parameters.

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An error analysis of the trace finite element discretization method applies to the surface finite element method of Dziuk-Elliott, resulting in optimal order discretized error bounds.

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