Question regarding diff operators for the HDivSurface space

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1

Dear NGSolvers,

A student and I are working on a (surface) shape optimization problem. As part of the problem we need to solve a linear elasticity problem on a surface, and as we might approach the incompressible case in our application, we want to employ a mixed formulation based on a the HDivSurface space plus weakly enforcement of the tangential continuity, very similar to what Wang/Ye did in their work (“New finite element methods in computational fluid dynamics by H (div) elements”, SINUM, 2007).

Therefore we are wondering how to correctly compute the surface divergence, and in particular the surface gradient and its correct surface transpose (or equivalently, the symmetric derivative ) for the test and trial functions from the HDivSurface space? Are these diff operators already implemented? Unfortunately, we haven’t managed to find any documentation regarding this space…

Best,
André

2

Hi Andre,

we can do H(div)-conforming Stokes on surfaces in NGSolve. The documentation on that is coming soon.

We prefer the version of Cockburn-Kanschat-Schötzau,
(A note on discontinuous Galerkin divergence-free solutions of the Navier–Stokes equations. Journal of Scientific Computing 2007),
in hybridized form as in our work with Christoph.

Is there an essential difference or advantage in Wang+Ye for you ?

Joachim

3

Hi Joachim,

and thanks for the status update. Yes, of course, a version of Cockburn-Kanschat-Schötzau would also work
for us, we were just simply looking for the “simplest” Hdiv conform approach to get started. Are there any estimates on when the HDivSurface related functionality will be released officially? Already looking forward to it!

Thanks and best,
Andre