@inproceedings{Hommel2000,
  author    = {Hommel, Angela},
  title     = {The Theory of Difference Potentials in the Three-Dimensional Case},
  doi       = {10.25643/bauhaus-universitaet.595},
  url       = {http://nbn-resolving.de/urn:nbn:de:gbv:wim2-20111215-5956},
  year      = {2000},
  abstract  = {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},
  subject      = {Randelemente-Methode},
  language  = {en}
}
@inproceedings{HommelRichter2003,
  author    = {Hommel, Angela and Richter, Matthias},
  title     = {Optimale Trassenf{\"u}hrung: Diskretisierung - Splineapproximation - Variationsmethoden},
  doi       = {10.25643/bauhaus-universitaet.309},
  url       = {http://nbn-resolving.de/urn:nbn:de:gbv:wim2-20111215-3094},
  year      = {2003},
  abstract  = {Ausgehend von mathematischen {\"U}berlegungen haben wir einfache Modellans{\"a}tze zur Bearbeitung des folgenden Optimierungsproblems erarbeitet und numerische Tests durchgef{\"u}hrt: Eine Landkarte wird in Quadrate unterteilt, wobei jedes Quadrat mit einem Faktor zu bewerten ist. Dieser Wichtungsfaktor sei klein, wenn das Gebiet problemlos passierbar ist und entsprechend groß, wenn es sich um ein Naturschutz-gebiet, einen See oder ein schwer befahrbares Gebiet handelt. Gesucht wird nach einer g{\"u}nstigen Verbindung vom Punkt A zum Punkt B, wobei die durch den Wichtungsfaktor gegebenen landschaftlichen Besonderheiten zu ber{\"u}cksichtigen sind. Wir formulieren das Problem zun{\"a}chst als Variationsproblem. Eine notwendige Bedingung, der die L{\"o}sungsfunktion gen{\"u}gen muß, ist die Euler-Lagrangesche Differentialgleichung. Mit Hilfe der Hamiltonschen Funktion ist es m{\"o}glich, diese Differentialgleichung in kanonischer Form zu schreiben. Durch Vereinfachung des Modelles gelingt es, das System der kanonischen Gleichungen so zu konkretisieren, daß es als Ausgangspunkt f{\"u}r numerische Untersuchungen betrachtet werden kann. Dazu verwandeln wir die urspr{\"u}ngliche Landschaft in eine >Berglandschaft<, wobei hohe Berge schwer passierbare Gebiete charakterisieren. Das einfachste Modell ist ein einzelner Berg, der mit Hilfe der Dichtefunktion einer zweidimensionalen Normalverteilung erzeugt wird. Zus{\"a}tzlich haben wir Berechnungen an zwei sich {\"u}berlagernden Bergen sowie einer Schlucht durchgef{\"u}hrt.},
  subject      = {Trassierung},
  language  = {de}
}
@inproceedings{Hommel2003,
  author    = {Hommel, Angela},
  title     = {Finite Difference Approximations of the Cauchy-Rieman Operators and the Solution of Discrete Stokes and Navier-Stokes Problems in the Plane},
  doi       = {10.25643/bauhaus-universitaet.307},
  url       = {http://nbn-resolving.de/urn:nbn:de:gbv:wim2-20111215-3073},
  year      = {2003},
  abstract  = {We give a summary of our results based on discrete Cauchy-Riemann operators in the plane. These operators are defined in a way that the factorization of the real Laplacian into two adjoint Cauchy-Riemann operators is possible. This property is similar to the continuous case and can especially be used for calculating the discrete fundamental solution of our finite difference operators. Based on the discrete fundamental solution we define a discrete operator that is right inverse to the discrete Cauchy-Riemann operator. In relation with this operator and an operator on the boundary we are able to prove a discrete version of the Borel-Pompeiu formula. In the second part we present a possibility to solve discrete Stokes and Navier-Stokes problems. The concept is based on the orthogonal decomposition of the space l2 into the space of discrete holomorphic functions and its orthogonal complement. By introducing the orthoprojectors P+h and Q+h we can prove the existence and uniqueness of the solution of discrete Stokes problems. In addition we state a problem that is equivalent to the discrete Navier-Stokes problem and can be used in an iteration procedure to describe the solution of this problem. For a special case of the Navier-Stokes equations we are able to calculate discrete potential and stream functions. The adapted model includes important algebraical properties and can immediately be used for numerical calculations. A numerical example is presented at the end of the article.},
  subject      = {Cauchy-Riemannsche Differentialgleichungen},
  language  = {en}
}