Partial differential equations on manifolds 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 manifolds have complicated structures. Traditional approaches require discretization on manifolds and projecting derivatives onto the tangent spaces of the manifolds. We introduce volumetric variational problems for solving such PDE's. We start with variational problems on manifolds and change them into extended problems in Eulerian formulation. The extended PDE's can be solved by many sophisticated numerical methods, such as finite element or finite difference methods. Based on special properties of the solutions, we design a special treatment for boundary conditions.
By Fourierand Laplace transformation, we analysis the method and show that it is stable for the elliptic and parabolictype of equations. However, it is not numerically stable for hyperbolic type equation. The instability can be fixed by modifying equations or reinitialization. We further study some nonconvex optimization problems: obstacle problems and TV-denoise problems on surfaces. Some numerical experiments are presented in the talk. This isa joint work with Richard Tsai.
Tamkang Journal of Mathematics started as a biannual journal in 1970 and is devoted to high-quality original research papers in pure and applied mathematics.