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Domain Decomposition Techniques

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Numerical Solution of Partial Differential Equations on Parallel Computers

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 51))

Summary

We introduce some parallel domain decomposition preconditioners for iterative solution of sparse linear systems like those arising from the approximation of partial differential equations by finite elements or finite volumes. We first give an overview of algebraic domain decomposition techniques. We then introduce a preconditioner based on a multilevel approximate Schur complement system. Then we present a Schwarz-based preconditioner augmented by an algebraic coarse correction operator. Being the definition of a coarse grid a difficult task on unstructured meshes, we propose a general framework to build a coarse operator by using an agglomeration procedure that operates directly on the matrix entries. Numerical results are presented aimed at assessing and comparing the effectiveness of the two methodologies. The main application will concern computational fluid dynamics (CFD), and in particular the simulation of compressible flow around aeronautical configurations.

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References

  1. J. Bramble, J. Pasciak, and X. Zhang. Two-level preconditioners for 2nd order elliptic finite element problems. East-West J. Numer. Math., 4:99–120, 1996.

    MathSciNet  Google Scholar 

  2. T. Chan, B. Smith, and J. Zou. Overlapping Schwarz methods on unstructured meshes using non-matching coarse grids. Numer. Math., 73:149–167, 1996.

    Article  MathSciNet  Google Scholar 

  3. T. Chan, S. Go, and J. Zou. Boundary treatments for multilevel methods on unstructured meshes. SIAM J. Sci. Comput., 21(1):46–66, 1999.

    Article  MathSciNet  Google Scholar 

  4. T. Chan and T. Mathew. The interface probing technique in domain decomposition. SIAM. Journal on Matrix Analysis and Applications, 13(1):212–238, 1992.

    Article  MathSciNet  Google Scholar 

  5. T. Chan and T. Mathew. Domain decomposition algorithms. Acta Numerica, pages 61–163, 1993.

    Google Scholar 

  6. T. Coffey, C. Kelley, and D. Keyes. Pseudotransient continuation and differential algebraic equations. SIAM Journal on Scientific Computing, 25(2):553–569, 1996.

    Article  MathSciNet  Google Scholar 

  7. H. Deconinck, H. Paillère, R. Struijs, and P. Roe. Multidimensional upwind schemes based on fluctuaction splitting for systems of conservation laws. Comput. Mech., 11:323–340, 1993.

    Article  Google Scholar 

  8. M. Dryja, B. Smith, and O. Widlund. Schwarz analysis of iterative substructuring algorithms for elliptic problems in three dimensions. SIAM J. Numer. Anal., 31(6):1662–1694, 1993.

    Article  MathSciNet  Google Scholar 

  9. M. Dryja and O. Widlund. Domain decomposition algorithms with small overlap. SIAM J. Sci. Comput., 15(3):604–620, 1994.

    Article  MathSciNet  Google Scholar 

  10. C. Farhat and F. Roux. A method of finite element tearing and interconnecting and its parallel solution algorithm. Internat. J. Numer. Meth. Engrg., 32:1205–1227, 1991.

    Article  Google Scholar 

  11. A. Grama, A. Gupta, and V. Kumar. Isoefficiency: Measuring the scalability of parallel algorithms and architectures. IEEE Parallel Distrib. Technol., 1:12–21, August 1993.

    Article  Google Scholar 

  12. F. Hülsemann, M. Kowarschik, M. Mohr, and U. Rüde. Parallel geometric multigrid. In A. M. Bruaset and A. Tveito, editors, Numerical Solution of Partial Differential Equations on Parallel Computers, volume 51 of Lecture Notes in Computational Science and Engineering, pages 165–208. Springer-Verlag, 2005.

    Google Scholar 

  13. K. Hwang. Advanced Computer Architecture: Parallelism Scalability, Programmability. McGraw Hill, New York, 1993.

    Google Scholar 

  14. K. Hwang and Z. Xu. Scalable Parallel Computing: Technology, Architecture, Programming. McGraw-Hill, Inc., New York, NY, USA, 1998.

    Google Scholar 

  15. G. Karypis and V. Kumar. METIS: Unstructured graph partitining and sparse matrix ordering system. Technical Report 98-036, University of Minnesota, Department of Computer Science, 1998.

    Google Scholar 

  16. P. Lin, M. Sala, J. Shadid, and R. Tuminaro. Performance of fully-coupled algebraic multilevel domain decomposition preconditioners for incompressible flow and transport. submitted to International Journal for Numerical Methods in Engineering, 2004.

    Google Scholar 

  17. A. Quarteroni and A. Valli. Domain Decomposition Methods for Partial Differential Equations. Oxford University Press, Oxford, 1999.

    Google Scholar 

  18. Y. Saad. Iterative Methods for Sparse Linear Systems. Thompson, Boston, 1996.

    Google Scholar 

  19. M. Sala. Amesos 2.0 reference guide. Technical Report SAND-4820, Sandia National Laboratories, September 2004.

    Google Scholar 

  20. M. Sala. Analysis of two-level domain decomposition preconditioners based on aggregation. Mathematical Modelling and Numerical Analysis, 38(5):765–780, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  21. B. Smith, P. Bjorstad, and W. Gropp. Domain Decomposition, Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press, New York, 1996.

    Google Scholar 

  22. X.-H. Sun. Scalability versus execution time in scalable systems. J. Parallel Distrib. Comput., 62(2):173–192, 2002.

    Article  MATH  Google Scholar 

  23. X.-H. Sun and D. Rover. Scalability of parallel algorithm-machine combinations. IEEE Parallel Distrib. Systems, 5:599–613, June 1994.

    Article  Google Scholar 

  24. P. L. Tallec. Domain decomposition methods in computational mechanics. Computational Mechanics Advances, 1:121–220, 1994.

    MATH  MathSciNet  Google Scholar 

  25. A. Toselli and O. Widlund. Domain Decomposition Methods-Algorithms and Theory, volume 34 of Springer Series in Computational Mathematics. Springer-Verlag, New York, 2005.

    Google Scholar 

  26. H. van der Vorst. Iterative Krylov Methods for Large Linear Systems, volume 13 of Cambridge. Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge, 2003.

    Google Scholar 

  27. H. van der Vorst and C. Vuik. GMRESR: a family of nested GMRES methods. Num. Lin. Alg. Appl., 1:369–386, 1994.

    Article  Google Scholar 

  28. P. Vanek, J. Mandel, and M. Brezina. Algebraic multigrid based on smoothed aggregation for second and fourth order problems. Computing, 56:179–196, 1996.

    Article  MathSciNet  Google Scholar 

  29. U. M. Yang. Parallel algebraic multigrid methods-high performance preconditioners. In A. M. Bruaset and A. Tveito, editors, Numerical Solution of Partial Differential Equations on Parallel Computers, volume 51 of Lecture Notes in Computational Science and Engineering, pages 209–236. Springer-Verlag, 2005.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Dipartimento di Matematica “F. Brioschi”, MOX, Politecnico di Milano, Italy

    Luca Formaggia & Fausto Saleri

  2. Sandia National Laboratories, Albuquerque, USA

    Marzio Sala

Authors
  1. Luca Formaggia
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  2. Marzio Sala
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  3. Fausto Saleri
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Editor information

Editors and Affiliations

  1. Simula Research Laboratory, P.O. Box 134, 1325, Lysaker, Fornebu, Norway

    Are Magnus Bruaset  & Aslak Tveito  & 

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Formaggia, L., Sala, M., Saleri, F. (2006). Domain Decomposition Techniques. In: Bruaset, A.M., Tveito, A. (eds) Numerical Solution of Partial Differential Equations on Parallel Computers. Lecture Notes in Computational Science and Engineering, vol 51. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-31619-1_4

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