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Convergence of the cotan Formula -an Overview Max Wardetzky

Summary: Convergence of the cotan Formula - an Overview
Max Wardetzky
Abstract. The cotan formula constitutes a discretization of the Laplace-Beltrami
operator on polyhedral surfaces in a Finite Element sense. In this note we give
an overview over its convergence properties. The mean curvature vector, given
by the Laplacian of the embedding of a surface, will serve as a case study: It
will be shown that mean curvature viewed as a functional converges, whereas
the corresponding piecewise linear functions do in general fail to converge.
1. Introduction
Discrete Differential Geometry is an emerging field which seeks discrete analogous
of classical differential-geometric concepts. Diverse approaches for discretization
have been brought forward over the past years - such as the theory of normal
cycles [6], the theory of circle patterns [2], and a geometric finite element (FEM)
theory on polyhedra [16]. This note gives an overview over convergence properties
of the FEM approach. The cotan formula, a widely used explicit expression for the
Laplace-Beltrami operator on polyhedral surfaces, serves as a prominent example.
A brief history of the cotan formula. The cotan representation for the Dirich-
let energy of piecewise linear functions on triangular nets seems to have first been
conceived by Duffin [8] in 1959. In 1988 Dziuk [9] studied a finite element (FEM)
approach for polyhedral surfaces - without stating this formula explicitly. In 1993


Source: Andrzejak, Artur - Konrad-Zuse-Zentrum für Informationstechnik Berlin


Collections: Computer Technologies and Information Sciences