A finite element approach for vector- and tensor-valued surface PDEs

TL;DR

This paper introduces a finite element method that reformulates vector- and tensor-valued surface PDEs into scalar PDEs using a covariant derivative approach, enabling the use of standard tools for their numerical solution.

Contribution

It provides a Cartesian componentwise formulation of covariant derivatives for tangential tensor fields, facilitating the solution of complex surface PDEs with established scalar PDE tools.

Findings

Achieved optimal linear convergence rates in numerical experiments.

Validated the approach on Helmholtz problems on an ellipsoid.

Demonstrated full functionality with a Landau-de Gennes problem on the Stanford bunny.

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.