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A Maximum Principle for L 2-Trace Norms with an Application to Optimized Schwarz Methods

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Domain Decomposition Methods in Science and Engineering XVIII

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 70))

Summary

Harmonic functions attain their pointwise maximum on the boundary of the domain. In this article, we analyze the relationship between various norms of nearly harmonic functions and we show that the trace norm is maximized on the boundary of the domain. One application is that the Optimized Schwarz Method with two subdomains converges for all Robin parameters α > 0.

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References

  1. Adams, R.A., Fournier, J.J.F.: Sobolev Spaces. Academic (Elsevier), Oxford, 2003.

    MATH  Google Scholar 

  2. Evans, L.C.: Partial Differential Equations. AMS, Providence, RI, 1998.

    MATH  Google Scholar 

  3. Gander, M.J.: Optimized Schwarz Methods. SIAM J. Numer. Anal., 44:699–731, 2006.

    Article  MATH  MathSciNet  Google Scholar 

  4. Kimn, J-H.: Overlapping Schwarz Algorithms using Discontinuous Iterates for Poisson’s Equation. Ph.D. Thesis, Department of Mathematics, New York University, New York, 2001.

    Google Scholar 

  5. Kimn, J.-H.: A Convergence Theory for An Overlapping Schwarz Algorithm using Discontinuous Iterates. Numer. Math., 100:117-139, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  6. Kufner, A.: Weighted Sobolev Spaces. Wiley, Chistester, U.K., 1985.

    MATH  Google Scholar 

  7. Lions, J.L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications I. Springer, 1970.

    Google Scholar 

  8. Lions, P.L.: On the Schwarz alternating method. III: A variant for nonoverlapping subdomains. In T. F. Chan, R. Glowinski, J. Périaux, O. Widlund, eds. Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, SIAM, 1990, Philadelphia, pp. 202–223.

    Google Scholar 

  9. Loisel, S., Szyld, D.B.: On the convergence of Algebraic Optimizable Schwarz Methods with applications to elliptic problems. Research Report 07-11-16, Department of Mathematics, Temple University, November 2007.

    Google Scholar 

  10. Nataf, F.: Absorbing boundary conditions in block Gauss-Seidel methods for convection problems. Math. Models Methods Appl. Sci., 6:481–502, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  11. Schwarz, H.: Über einige Abbilaungsaufgaben. J. Reine Angew. Math., vol. 70, 105–120, 1869.

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Temple University, Philadelphia, PA, 19122-6094, USA

    Sébastien Loisel & Daniel B. Szyld

Authors
  1. Sébastien Loisel
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  2. Daniel B. Szyld
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Corresponding author

Correspondence to Sébastien Loisel .

Editor information

Editors and Affiliations

  1. School of Computer Science and Engineering, The Hebrew University of Jerusalem, Ross Building - Givat Ram Campus, 91904, Jerusalem, Israel

    Michel Bercovier

  2. Section de mathématiques, Université de Genève, 2-4 rue du Lièvre, 1211, Genève 24, Switzerland

    Martin J. Gander

  3. Institut für Mathematik, Freie Universität Berlin, Arnimallee 6, 14195, Berlin-Dahlem, Germany

    Ralf Kornhuber

  4. Courant Institute, 251 Mercer Street, New York, NY, 10012, USA

    Olof Widlund

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© 2009 Springer-Verlag Berlin Heidelberg

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Loisel, S., Szyld, D.B. (2009). A Maximum Principle for L 2-Trace Norms with an Application to Optimized Schwarz Methods. In: Bercovier, M., Gander, M.J., Kornhuber, R., Widlund, O. (eds) Domain Decomposition Methods in Science and Engineering XVIII. Lecture Notes in Computational Science and Engineering, vol 70. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02677-5_20

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