AFEM for geometric PDE: The Laplace-Beltrami operator

Abstract

We present several applications governed by geometric PDE, and their parametric finite element discretization, which might yield singular behavior. The success of such discretization hinges on an adequate variational formulation of the Laplace-Beltrami operator, which we describe in detail for polynomial degree 1. We next present a complete a posteriori error analysis which accounts for the usual PDE error as well as the geometric error induced by interpolation of the surface. This leads to an adaptive finite element method (AFEM) and its convergence. We discuss a contraction property of AFEM and show its quasi-optimal cardinality.