@article{LegatiukGuerlebeckHommel,
  author    = {Legatiuk, Anastasiia and G{\"u}rlebeck, Klaus and Hommel, Angela},
  title     = {Estimates for the discrete fundamental solution of the discrete Laplace operator on a rectangular lattice},
  series = {Mathematical Methods in the Applied Sciences},
  volume    = {2021},
  journal   = {Mathematical Methods in the Applied Sciences},
  publisher = {Wiley},
  address   = {Chichester},
  doi       = {10.1002/mma.7747},
  url       = {http://nbn-resolving.de/urn:nbn:de:gbv:wim2-20220209-45829},
  pages     = {1 -- 23},
  abstract  = {This paper presents numerical analysis of the discrete fundamental solution of the discrete Laplace operator on a rectangular lattice. Additionally, to provide estimates in interior and exterior domains, two different regularisations of the discrete fundamental solution are considered. Estimates for the absolute difference and lp-estimates are constructed for both regularisations. Thus, this work extends the classical results in the discrete potential theory to the case of a rectangular lattice and serves as a basis for future convergence analysis of the method of discrete potentials on rectangular lattices.},
  subject      = {diskrete Fourier-Transformation},
  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{HommelGuerlebeck,
  author    = {Hommel, Angela and G{\"u}rlebeck, Klaus},
  title     = {THE RELATIONSHIP BETWEEN LINEAR ELASTICITY THEORY AND COMPLEX FUNCTION THEORY STUDIED ON THE BASIS OF FINITE DIFFERENCES},
  series = {Digital Proceedings, International Conference on the Applications of Computer Science and Mathematics in Architecture and Civil Engineering : July 20 - 22 2015, Bauhaus-University Weimar},
  booktitle = {Digital Proceedings, International Conference on the Applications of Computer Science and Mathematics in Architecture and Civil Engineering : July 20 - 22 2015, Bauhaus-University Weimar},
  editor    = {G{\"u}rlebeck, Klaus and Lahmer, Tom},
  organization    = {Bauhaus-Universit{\"a}t Weimar},
  issn      = {1611-4086},
  doi       = {10.25643/bauhaus-universitaet.2801},
  url       = {http://nbn-resolving.de/urn:nbn:de:gbv:wim2-20170314-28010},
  pages     = {6},
  abstract  = {It is well-known that the solution of the fundamental equations of linear elasticity for a homogeneous isotropic material in plane stress and strain state cases can be equivalently reduced to the solution of a biharmonic equation. The discrete version of the Theorem of Goursat is used to describe the solution of the discrete biharmonic equation by the help of two discrete holomorphic functions. In order to obtain a Taylor expansion of discrete holomorphic functions we introduce a basis of discrete polynomials which fulfill the so-called Appell property with respect to the discrete adjoint Cauchy-Riemann operator. All these steps are very important in the field of fracture mechanics, where stress and displacement fields in the neighborhood of singularities caused by cracks and notches have to be calculated with high accuracy. Using the sum representation of holomorphic functions it seems possible to reproduce the order of singularity and to determine important mechanical characteristics.},
  subject      = {Angewandte Informatik},
  language  = {en}
}
@phdthesis{Hommel,
  author    = {Hommel, Angela},
  title     = {Diskret holomorphe Funktionen und deren Bedeutung bei der L{\"o}sung von Differenzengleichungen},
  doi       = {10.25643/bauhaus-universitaet.3784},
  url       = {http://nbn-resolving.de/urn:nbn:de:gbv:wim2-20180827-37846},
  school      = {Bauhaus-Universit{\"a}t Weimar},
  abstract  = {Auf der Grundlage diskreter Cauchy-Riemann Operatoren werden diskret holomorphe Funktionen definiert und detailliert studiert. Darauf aufbauend wird die L{\"o}sung von Differenzengleichungen mit Hilfe der diskret holomorphen Funktionen beschrieben.},
  subject      = {Differenzengleichung},
  language  = {de}
}
@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}
}
@phdthesis{Hommel1998,
  author    = {Hommel, Angela},
  title     = {Fundamentall{\"o}sungen partieller Differenzenoperatoren und die L{\"o}sung diskreter Randwertprobleme mit Hilfe von Differenzenpotentialen},
  doi       = {10.25643/bauhaus-universitaet.28},
  url       = {http://nbn-resolving.de/urn:nbn:de:gbv:wim2-20040216-303},
  school      = {Bauhaus-Universit{\"a}t Weimar},
  year      = {1998},
  abstract  = {Im Mittelpunkt der Dissertation steht die Theorie der Differenzenpotentiale, die eng mit der klassischen Potentialtheorie verbunden ist. Vorgestellt wird eine Methode zur L{\"o}sung von Randwertproblemen, die nicht auf der Diskretisierung einer Randintegralgleichung beruht, sondern von der {\"U}bertragung des Problems in ein Differenzenrandwertproblem ausgeht. Das diskrete Randwertproblem wird mit Hilfe einer Randreduktionsmethode auf eine Randoperatorgleichung transformiert, die detaillierter zu untersuchen ist. Voraussetzung f{\"u}r den Aufbau der Theorie ist die Existenz diskreter Fundamentall{\"o}sungen. Die Definition der Differenzenpotentiale wird von Ryabenkij {\"u}bernommen. Seine Herangehensweise f{\"u}hrt jedoch zu {\"u}berbestimmten linearen Gleichungssystemen auf dem Rand. Durch die Aufspaltung des Randpotentials in ein diskretes Einfach- und Doppelschichtpotential wird diese Schwierigkeit in der Dissertation {\"u}berwunden. Bewiesen werden Eindeutigkeits- und L{\"o}sbarkeitsaussagen f{\"u}r Differenzenrandwertprobleme. Das onvergenzverhalten der diskreten Potentiale wird im Kapitel 3 untersucht. Im Kapitel 4 werden numerische Resultate vorgestellt.},
  subject      = {Diskrete Fourier-Transformation},
  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}
}