I will discuss two recent approaches to constructing intrinsic finite element spaces for tensor fields, developed jointly with Evan Gawlik.
One approach, blow-up finite elements, was motivated by a vexing problem when discretizing tangent vector fields on surfaces: With a polyhedral discretization of the surface, the angles at a vertex no longer sum to 360 degrees; as a result, it is not possible to construct a vector field that is continuous within each element, tangent to the surface, and continuous across each edge (in the sense of having no jump in the components tangent to the edge and normal to the edge). Previous approaches either broke tangentiality to the surface or continuity across edges. With blow-up finite elements, we can keep both of these properties by allowing the vector fields to vary rapidly near vertices. I will define these elements for vector fields and tensor fields, discuss some preliminary numerical results, and discuss potential applications to numerical geometry and to intrinsic discretization of the surface Stokes equations for creeping flow.
The second approach was motivated by recent interest in extending finite element exterior calculus from differential forms to double forms, also known as form-valued forms, for applications to elasticity and numerical geometry. A key property of finite element exterior calculus spaces is their invariance under affine transformations, which, in particular, means that the spaces work the same way on surfaces as they do in the plane. I will discuss our construction of affine-invariant spaces of double forms with polynomial coefficients, the surprising connections to representation theory, and ongoing work on generalizing these results to arbitrary covariant tensor fields.