Foundations Of Computational Mathematics (FoCM 2005)

Workshop on Foundations of Numerical PDEs

July 7 - July 9, 2005

Universidad de Cantabria in Santander, Spain

Finite Element Methods for Moving Surfaces and Applications to Stressed Epitaxial Films, Shape Optimization, and Image Processing

Ricardo Nochetto

Mathematics and IPST at University of Maryland

Abstract:   Using shape differential calculus, which expresses variations of bulk and surface energy with respect to domain changes, and Euler implicit time discretization, we formulate gradient flows for these energies which yield geometric laws for the motion of domain boundaries (curves or surfaces). We next present a semi-implicit variational formulation which requires no explicit parametrization of the surface, and is sufficiently flexible to accomodate several scalar products for the computation of normal velocity, depending on the application. This leads to linear systems of lower order elliptic PDE to solve at each time step, in both the surface and bulk. We develop adaptive finite element methods (AFEM), and propose a Schur complement approach to solve the resulting linear SPD systems. We first apply this idea to surface diffusion, namely to the geometric motion of a surface with normal velocity proportional to the surface Laplacian of mean curvature, and couple it with elasticity in the bulk; this is a simple model for stressed epitaxial films. We present several numerical experiments for surface diffusion including pinch-off in finite time and topological changes. We also present preliminary computations for the coupled system exhibiting formation of dislocations. We next discuss applications to shape optimization and image processing, that is to the minimization of functionals subject to differential constraints, and present preliminary simulations. We discuss time and space adaptivity to handle the multiscale nature of these problems, as well as mesh generation, mesh distortion and mesh smoothing. This work is joint with E. Baensch, G. Dogan, P. Morin, and M. Verani.