This thesis studies discrete differential-geometric analogues of the smooth theory of two-dimensional Riemannian manifolds in the framework of discrete differential geometry (DDG). In particular, the following objects are considered on polyhedral surfaces: Laplace-Beltrami operator, gradient, divergence, curl, solutions to the Dirichlet problem, mean curvature vector, geodesics, complex structure, de Rham cohomology, Hodge decomposition, Hodge star operator, and spectrum of the Laplace operator. The discretization of these objects is primarily built upon the discretization of function spaces on polyhedra in the sense of linear finite elements. Accordingly, the first part of this thesis discusses weak derivatives and Sobolev spaces on polyhedral surfaces from which discrete differential complexes and their cohomological properties are derived. The second part deals with convergence and approximation properties of discrete differential operators. In particular, it is shown that the aforementioned (discrete) objects converge to their smooth counterparts if the points and normals of a sequence of polyhedral surfaces embedded into Euclidean 3-space converge to those of a smooth limit surface. A particular emphasis is put on the appropriate norms in which convergence can be expected. Several applications, specifically to computer graphics, are mentioned along the way.
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