A finite element approach for vector- and tensor-valued surface PDEs
This work provides a practical framework for solving vector- and tensor-valued surface PDEs using existing scalar solvers, which is useful for researchers in computational geometry and materials science.
The authors derive a Cartesian componentwise description of covariant derivatives for tangential tensor fields, enabling the use of standard scalar-valued surface PDE solvers for vector- and tensor-valued problems. They demonstrate optimal linear convergence for Helmholtz problems on an ellipsoid and apply the method to a Landau-de Gennes problem on the Stanford bunny.
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.