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Discrete Differential Operators on Polyhedral Surfaces - Convergence and Approximation
Wardetzky, Max

HaupttitelDiscrete Differential Operators on Polyhedral Surfaces - Convergence and Approximation
TitelvarianteDiskrete Differentialoperatoren auf Polyedrischen Flächen - Konvergenz und Approximation
AutorWardetzky, Max
Geburtsort: Berlin, Deutschland
GutachterProf. Dr. Konrad Polthier
weitere GutachterProf. Dr. Alexander I. Bobenko
Prof. Dr. Jean-Marie Morvan
Freie Schlagwörterdiscrete differential geometry, convergence, finite elements, discrete Laplacian, discrete mean curvature, discrete Hodge star52B70
DDC510 Mathematik
ZusammenfassungThis 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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Fachbereich/EinrichtungFB Mathematik und Informatik
Erscheinungsjahr2007
Dokumententyp/-SammlungenDissertation
Medientyp/FormatText
SpracheEnglisch
RechteNutzungsbedingungen
Tag der Disputation08.11.2006
Erstellt am05.10.2007 - 00:00:00
Letzte Änderung19.02.2010 - 14:00:17
 
Alte Darwin URLhttp://www.diss.fu-berlin.de/2007/663/
Statische URLhttp://www.diss.fu-berlin.de/diss/receive/FUDISS_thesis_000000003188
URNurn:nbn:de:kobv:188-fudissthesis000000003188-7
Zugriffsstatistik