Speaker
Description
Blow-up finite elements, developed jointly with Evan Gawlik, were motivated by a vexing problem when discretizing tangent vector fields on surfaces: For a discretized surface, the angles at vertices generally no longer sum to 360 degrees. As a result, it is not possible to construct a vector field approximation that is continuous within each element, tangent to the surface, and continuous across each edge (in the sense that there is no jump in the component tangent to the edge and no jump in the component 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. The resulting vector fields are both tangent to the surface and single-valued on edges, but they are multi-valued at 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.