Richard Tsai: Volumetric Variational Problems for a Class of Partial Differential Equations on Surfaces

Abstract:

Partial differential equations on surfaces are important for many areas, such as materials science, fluid dynamics, and biology. The computation of such problems can be costly and difficult when the surface has complicated structures. Our aim is to develop a new formulation which bypasses the need of parametrization of surfaces and which allows  for a wide variety of numerical methods to be implemented. We consider surface PDEs which can be derived from minimization of some energies defined on the surfaces. We introduce volumetric variational problems for solving such PDE’s in a thin narrowband around the surfaces. We start with variational problems on surfaces and change them into volumetric variational problems in an Eulerian formulation. For energies depending on first derivatives of the unknowns, the resulted PDE's are standard elliptic type equations. We further exploit a special property of the solutions to implement the boundary conditions.  Some numerical experiments are presented in the talk.