Topic 2

Ricardo H. Nochetto. AFEM for the Laplace-Beltrami Operator on Surfaces.

ABSTRACT: We first introduce an AFEM for the Laplace-Beltrami operator on C1 graphs in R^d (d>=2). We derive a posteriori error estimates that account for boththe energy error in H1 and the geometric error in W^{1,1} due to approximation of the surface by a polyhedral one. We devise a marking strategy to reduce the energyand geometric errors as well as the geometric oscillation. We prove that AFEM is a contraction on a suitably scaled sum of these three quantities as soon as the geometric oscillation has been reduced beyond a threshold. The resulting AFEM converges without knowing such threshold or any constants, and starting from anycoarse initial triangulation. Several numerical experiments illustrate the theory. Finally, we introduce and analyze an AFEM for the Laplace-Beltrami operatoron parametric surfaces, thereby extending the results for graphs. Note that, due to the nature of parametric surfaces, the geometric oscillation is now measured in terms of the differences of tangential gradients rather than differences of normals as for graphs. Numerical experiments with closed surfaces are provided to illustrate the theory.