Discrete Differential Geometry
Discrete Differential Geometry
Convergence of the Cotangent Formula: An Overview (p. 275-276)
Abstract. The cotangent formula constitutes an intrinsic discretization of the Laplace- Beltrami operator on polyhedral surfaces in a finite-element sense. This note gives an overview of approximation and convergence properties of discrete Laplacians and mean curvature vectors for polyhedral surfaces located in the vicinity of a smooth surface in euclidean 3-space. In particular, we show that mean curvature vectors converge in the sense of distributions, but fail to converge in L2. Keywords. Cotangent formula, discrete Laplacian, Laplace-Beltrami operator, convergence, discrete mean curvature.
There are various approaches toward a purely discrete theory of surfaces for which classical differential geometry, and in particular the notion of curvature, appears as the limit case. Examples include the theory of spaces of bounded curvature [1, 24], Lipschitz- Killing curvatures [5, 12, 13], normal cycles [6, 7, 30, 31], circle patterns and discrete conformal structures [2, 17, 26, 28], and geometric finite elements [10, 11, 15, 20, 29]. In this note we take a finite-element viewpoint, or, more precisely, a functional-analytic one, and give an overview over convergence properties of weak versions of the Laplace- Beltrami operator and the mean curvature vector for embedded polyhedral surfaces. Convergence. Consider a sequence of polyhedral surfaces fMng, embedded into euclidean 3-space, which converges (in an appropriate sense) to a smooth embedded surface M. One may ask: What are the measures and conditions such that metric and geometric objects on Mn-like intrinsic distance, area, mean curvature, Gauss curvature, geodesics and the Laplace-Beltrami operator-converge to the corresponding objects on M? To date no complete answer has been given to this question in its full generality. For example, the approach of normal cycles [6, 7], while well-suited for treating convergence of curvatures of embedded polyhedra in the sense of measures, cannot deal with convergence of elliptic operators such as the Laplacian. The finite-element approach, on the other hand, while well-suited for treating convergence of elliptic operators (cf. [10, 11]) and mean curvature vectors, has its difficulties with Gauss curvature.
Despite the differences between these approaches, there is a remarkable similarity: The famous lantern of Schwarz  constitutes a quite general example of what can go wrong-pointwise convergence of surfaces without convergence of their normal fields. Indeed, while one cannot expect convergence of metric and geometric properties of embedded surfaces from pointwise convergence alone, it often suffices to additionally require convergence of normals. The main technical step, to show that this is so, is the construction of a bi-Lipschitz map between a smooth surface M, embedded into euclidean 3- space, and a polyhedral surfaceMh nearby, such that the metric distortion induced by this map is bounded in terms of the Hausdorff distance between M and Mh, the deviation of normals, and the shape operator of M. (See Theorem 3.3 and compare  for a similar result.) This map then allows for explicit error estimates for the distortion of area and length, and-when combined with a functional-analytic viewpoint-error estimates for the Laplace-Beltrami operator and the mean curvature vector.
We treat convergence of Laplace-Beltrami operators in operator norm, and we discuss two distinct concepts of mean curvature: a functional representation (in the sense of distributions) as well as a representation as a piecewise linear function. We observe that one concept (the functional) converges whereas the other (the function) in gen