TY  - THES
A1  - Hommel, Angela
T1  - Fundamentallösungen partieller Differenzenoperatoren und die Lösung diskreter Randwertprobleme mit Hilfe von Differenzenpotentialen
T1  - Fundamental Solutions for Partial Difference Operators and the Solution of Discrete Boundary Value Problems by the Help of Difference Potentials
N2  - Im Mittelpunkt der Dissertation steht die Theorie der Differenzenpotentiale, die eng mit der klassischen Potentialtheorie verbunden ist. Vorgestellt wird eine Methode zur Lösung von Randwertproblemen, die nicht auf der Diskretisierung einer Randintegralgleichung beruht, sondern von der Übertragung des Problems in ein Differenzenrandwertproblem ausgeht. Das diskrete Randwertproblem wird mit Hilfe einer Randreduktionsmethode auf eine Randoperatorgleichung transformiert, die detaillierter zu untersuchen ist. Voraussetzung für den Aufbau der Theorie ist die Existenz diskreter Fundamentallösungen. Die Definition der Differenzenpotentiale wird von Ryabenkij übernommen. Seine Herangehensweise führt jedoch zu überbestimmten linearen Gleichungssystemen auf dem Rand. Durch die Aufspaltung des Randpotentials in ein diskretes Einfach- und Doppelschichtpotential wird diese Schwierigkeit in der Dissertation überwunden. Bewiesen werden Eindeutigkeits- und Lösbarkeitsaussagen für Differenzenrandwertprobleme. Das onvergenzverhalten der diskreten Potentiale wird im Kapitel 3 untersucht. Im Kapitel 4 werden numerische Resultate vorgestellt.
N2  - The theses are based on the theory of difference potentials, which are closely related to the classical potential theory. A method for solving boundary value problems is presented, that does not start from the discretization of a boundary integral equation. In the first step the original problem is replaced by a discrete boundary value problem. By the help of a boundary reduction method the discrete problem is transformed into a boundary operator equation, which is to study in more detail. An important assumption for the theory of difference potentials is the existence of discrete fundamental solutions. The definition of the difference potentials is taken from Ryabenkij. His approach leads to overdetermined linear equation systems on the boundary. By splitting the boundary potential into a discrete single-layer and double-layer potential these problems are solved in the theses. Uniqueness and existence theorems are proved for discrete boundary value problems. The convergence of the discrete potentials is investigated in chapter 3. In chapter 4 numerical results are presented.
KW  - Diskrete Fourier-Transformation
KW  - Randwertproblem
KW  - Greensche Matrix
KW  - diskrete Fundamentallösung
KW  - Lösung innerer und äußerer Randwertprobleme
KW  - Differenzenpotentiale
KW  - diskretes Einfach- und Doppelschichtpotential
KW  - discrete Fourier transform
KW  - discrete fundamental solution
KW  - solution of inner and outer boundary value problems
KW  - difference potentials
Y1  - 1998
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20040216-303
ER  - 
TY  - CHAP
A1  - Hommel, Angela
T1  - The Theory of Difference Potentials in the Three-Dimensional Case
N2  - The method of difference potentials can be used to solve discrete elliptic boundary value problems, where all derivatives are approximated by finite differences. Considering the classical potential theory, an integral equation on the boundary will be investigated, which is solved approximately by the help of a quadrature formula. The advantage of the discrete method consists in the establishment of a linear equation system on the boundary, which can be immediately solved on the computer. The described method of difference potentials is based on the discrete Laplace equation in the three-dimensional case. In the first step the integral representation of the discrete fundamental solution is presented and the convergence behaviour with respect to the continuous fundamental solution is discussed. Because the method can be used to solve boundary value problems in interior as well as in exterior domains, it is necessary to explain some geometrical aspects in relation with the discrete domain and the double-layer boundary. A discrete analogue of the integral representation for functions in will be presented. The main result consists in splitting the difference potential on the boundary into a discrete single- and double-layer potential, respectively. The discrete potentials are used to establish and solve a linear equation system on the boundary. The actual form of this equation systems and the conditions for solvability are presented for Dirichlet and Neumann problems in interior as well as in exterior domains
KW  - Randelemente-Methode
KW  - Diskrete Fourier-Transformation
Y1  - 2000
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20111215-5956
ER  -