Finite Difference Approximations of the Cauchy-Rieman Operators and the Solution of Discrete Stokes and Navier-Stokes Problems in the Plane

  • 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 theWe 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.show moreshow less

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Metadaten
Document Type:Conference Proceeding
Author: Angela HommelGND
DOI (Cite-Link):https://doi.org/10.25643/bauhaus-universitaet.307Cite-Link
URN (Cite-Link):https://nbn-resolving.org/urn:nbn:de:gbv:wim2-20111215-3073Cite-Link
Language:English
Date of Publication (online):2005/01/04
Year of first Publication:2003
Release Date:2005/01/04
Institutes and partner institutions:Fakultät Bauingenieurwesen / Professur Informatik im Bauwesen
GND Keyword:Cauchy-Riemannsche Differentialgleichungen; Navier-Stokes-Gleichung
Source:Internationales Kolloquium über Anwendungen der Informatik und Mathematik in Architektur und Bauwesen , IKM , 16 , 2003 , Weimar , Bauhaus-Universität
Dewey Decimal Classification:600 Technik, Medizin, angewandte Wissenschaften / 620 Ingenieurwissenschaften / 620 Ingenieurwissenschaften und zugeordnete Tätigkeiten
BKL-Classification:31 Mathematik / 31.80 Angewandte Mathematik
56 Bauwesen / 56.03 Methoden im Bauingenieurwesen
Collections:Bauhaus-Universität Weimar / Internationales Kolloquium über Anwendungen der Informatik und Mathematik in Architektur und Bauwesen, IKM, Weimar / Internationales Kolloquium über Anwendungen der Informatik und Mathematik in Architektur und Bauwesen, IKM, Weimar, 16. 2003
Licence (German):License Logo In Copyright