A finite element approach for vector- and tensor-valued surface PDEs
Abstract
We derive a Cartesian componentwise description of the covariant derivative of tangential tensor fields of any degree on general manifolds. This allows to reformulate any vector- and tensor-valued surface PDE in a form suitable to be solved by established tools for scalar-valued surface PDEs. We consider piecewise linear Lagrange surface finite elements on triangulated surfaces and validate the approach by a vector- and a tensor-valued surface Helmholtz problem on an ellipsoid. We experimentally show optimal (linear) order of convergence for these problems. The full functionality is demonstrated by solving a surface Landau-de Gennes problem on the Stanford bunny. All tools required to apply this approach to other vector- and tensor-valued surface PDEs are provided.
Cite
@article{arxiv.1809.00945,
title = {A finite element approach for vector- and tensor-valued surface PDEs},
author = {Michael Nestler and Ingo Nitschke and Axel Voigt},
journal= {arXiv preprint arXiv:1809.00945},
year = {2019}
}