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Zusammenfassung

Diskretisierungen elliptischer und parabolischer Randwertprobleme führen auf äuβerst dünn besetzte lineare Gleichungssysteme. Diese besitzen zwar Bandstruktur, sind jedoch gewöhnlich auch innerhalb des Bandes nur sehr spärlich besetzt. Mit Ausnahme der Multisplitting-Verfahren sind daher die bisher besprochenen parallelen Verfahren für solche in der numerischen Praxis besonders wichtigen Gleichungssysteme nicht geeignet.

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References

  1. Adams, L., Jordan, H.: Is SOR Color-blind?, SIAM J. Sci. Stat. Comput. 7, 490–501 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  2. Adams, L., LeVeque, R., Young, D.: Analysis of the SOR Iteration for the 9-Point Laplacian, SIAM J. Numer. Anal. 25, 1156–1180 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  3. Adams, L., Ortega, J.: A Multicolour SOR Method for Parallel Computation, Proc. of the 1982 International Conference on Parallel Processing, Bellaire, MI, August 1982, 53–56

    Google Scholar 

  4. Axelsson, O., Barker, V.: Finite Element Solution of Boundary Value Problems, New York: Academic Press (1984)

    MATH  Google Scholar 

  5. Block, U., Frommer, A., Mayer, G.: Block Colouring Schemes for the SOR Method on Local Memory Parallel Computers, erscheint in Parallel Comput. (1990)

    Google Scholar 

  6. Buneman, O.: A Compact Non-iterative Poisson Solver, Stanford University, Institute for Plasma Research Report Nr. 294, Stanford CA (1969)

    Google Scholar 

  7. Buzbee, B., Golub, G., Nielson, C.: On Direct Methods for Solving Poisson’s Equations, SIAM J. Numer. Anal. 7, 627–656 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  8. Collatz, L.: The Numerical Treatment of Differential Equations, 3rd Edition, Berlin: Springer (1960)

    MATH  Google Scholar 

  9. Dorr, F.: The Direct Solution of the Discrete Poisson Equation on a Rectangle, SIAM Rev. 12, 248–2263 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  10. Duff, I., Meurant, G.: The Effect of Ordering on Preconditioned Conjugate Gradient Methods, Technical Report HL88/1414, Computer Science and Systems Division, Harwell Laboratory, Oxon OX11 0RA (1988)

    Google Scholar 

  11. Eiermann, M., Varga, R., Niethammer, W.: Iterationsverfahren für nichtsymmetrische Gleichungssysteme und Approximationsmethoden im Komplexen, Jahresb. Dtsch. Math.-Ver. 89, 1–32 (1987)

    MATH  MathSciNet  Google Scholar 

  12. Gentzsch, W.: A Fully Vectorizable SOR Variant, Parallel Comput. 4, 349–353 (1987)

    Article  MATH  Google Scholar 

  13. Gesellschaft für Mathematik und Datenverarbeitung (Hrsg.): GMD-Spiegel 19, Heft 1, Sankt Augustin: GMD (1989)

    Google Scholar 

  14. Giloi, W.: SUPRENUM: A Trendsetter in Modern Supercomputer Development, Parallel Comput. 7, 283–296 (1988)

    Article  Google Scholar 

  15. Golub, G., van Loan, Ch.: Matrix Computations, 2nd Edition, Baltimore: Johns Hopkins (1989)

    MATH  Google Scholar 

  16. Hackbusch, W.: Multi-Grid Methods and Applications, Berlin: Springer (1985)

    MATH  Google Scholar 

  17. Hockney, R., Jesshope, C.: Parallel Computers 2, Bristol: Adam Hilger (1988)

    MATH  Google Scholar 

  18. Kincaid, D., Oppe, T., Young, D.: Vector Computations for Sparse Linear Systems, SIAM J. Alg. Disc. Meth. 7, 99–112 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  19. Kincaid, D., Oppe, T., Young, P.: Vectorized Iterative Methods for Partial Differential Equations, Commun. Appl. Numer. Math. 2, 789–796 (1986)

    Google Scholar 

  20. LeVeque, R., Trefethen, L.: Fourier Analysis of the SOR Iteration, IMA J. Numer. Anal. 8, 273–279 (1988)

    Article  MathSciNet  Google Scholar 

  21. Meis, Th., Marcowitz, U.: Numerische Behandlung von Differentialgleichungen, Berlin: Springer (1978)

    Google Scholar 

  22. Ortega, J.: Introduction to Parallel and Vector Solution of Linear Systems, New York: Plenum Press (1988)

    MATH  Google Scholar 

  23. Ortega, J., Voigt, R.: Solution of Partial Differential Equations on Vector and Parallel Computers, SIAM Rev. 27, 149–270 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  24. Press, W., Flannery, B., Teukolsky, S., Vetterling, W.: Numerical Recipes, Cambridge: Cambridge University Press (1986)

    Google Scholar 

  25. Reichel, L.: The Ordering of Tridiagonal Matrices in the Cyclic Reduction Method for Poisson’s Equation, Numer. Math. 56, 215–227 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  26. Saad, Y.: Krylov Subspace Methods on Supercomputers, SIAM J. Sci. Stat. Comput. 10, 1200–1232 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  27. Saltz, J., Naik, V., Nicol, D.: Reduction of the Effects of the Communication Delays in Scientific Algorithms on Message Passing MIMD Architectures, SIAM J. Sci. Stat. Comput. 8, s118–s138 (1987)

    Article  Google Scholar 

  28. Schönauer, W.: Scientific Computation on Vector Computers, Amsterdam: North Holland (1987)

    Google Scholar 

  29. Schwarz, H.: Numerische Mathematik, Stuttgart: Teubner (1986)

    MATH  Google Scholar 

  30. Solchenbach, K.: Grid Applications on Distributed Memory Architectures: Implementation and Evaluation, Parallel Comput. 7, 341–356 (1988)

    Article  MATH  Google Scholar 

  31. Solchenbach, K., Trottenberg, U.: SUPRENUM: System Essentials and Grid Applications, Parallel Comput.7, 265–281 (1988)

    Article  MATH  Google Scholar 

  32. Stoer, J., Bulirsch, R.: Einführung in die Numerische Mathematik II, 2. Auflage, Berlin: Springer (1978)

    MATH  Google Scholar 

  33. Trottenberg, U.: SUPRENUM — an MIMD Multiprocessor System for Multi-Level Scientific Computing, in: Händler, W. et al.(eds.): CONPAR 86, Lecture Notes in Computer Science 237, 48–52 (1986)

    Google Scholar 

  34. Varga, R.: Matrix Iterative Analysis, Englewood Cliffs: Prentice Hall (1962)

    Google Scholar 

  35. van der Vorst, H.: High Performance Preconditioning, SIAM J. Sci. Stat. Corn-put. 10, 1174–1185 (1989)

    Article  MATH  Google Scholar 

  36. Young, D.: Iterative Solution of Large Linear Systems, New York: Academic Press (1971)

    MATH  Google Scholar 

Download references

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© 1990 Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig

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Frommer, A. (1990). Modellproblem: Diskrete Laplace—Gleichung. In: Lösung linearer Gleichungssysteme auf Parallelrechnern. Vieweg+Teubner Verlag. https://doi.org/10.1007/978-3-322-83922-0_10

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  • DOI: https://doi.org/10.1007/978-3-322-83922-0_10

  • Publisher Name: Vieweg+Teubner Verlag

  • Print ISBN: 978-3-528-06397-9

  • Online ISBN: 978-3-322-83922-0

  • eBook Packages: Springer Book Archive

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