 
Summary: Convergence of the cotan Formula  an Overview
Max Wardetzky
Abstract. The cotan formula constitutes a discretization of the LaplaceBeltrami
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 differentialgeometric 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
LaplaceBeltrami 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
