Abstract
By using interval techniques, it is possible to obtain global convergence properties and verified enclosures in the numerical solution of several classes of nonlinear systems of equations. In the present paper, we introduce Newton-like interval methods of the so-called Krawczyk-type for systems arizing from discretizations of almost linear parabolic problems. Parallelism is introduced by domain decomposition and an adaptation of the Schwarz Alternating Procedure. Numerical results from a Sun Opteron cluster are included.
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References
Alefeld, G., Herzberger, J.: Introduction to Interval Computations. Academic Press, New York (1983)
Frommer, A., Schwandt, H.: A Unified Representation and Theory of Algebraic Additive Schwarz and Multisplitting Methods. SIAM J. Matrix Anal. Appl. 15, 893–912 (1997)
Krawczyk, R.: Newton–Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken. Computing 4, 187–201 (1969)
Ortega, J., Rheinboldt, W.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1970)
Rodrigue, G.: Inner/outer iterative methods and numerical Schwarz algorithms. Parallel Comp. 2, 205–218 (1985)
Rodrigue, G., Shan, K.L., Yu-Hui, L.: Convergence and comparison analysis of some numerical Schwarz methods. Numer. Math. 56, 123–138 (1989)
Schwandt, H.: Krawczyk-like Algorithms for the Solution of Systems of Nonlinear Equations. SIAM J. Num. Anal. 22, 792–810 (1985)
Schwandt, H.: The Solution of Nonlinear Elliptic Dirichlet Problems on Rectangles by Almost Globally Convergent Interval Methods. SIAM J. Sc. St. Comp. 6, 617–638 (1985)
Varga, R.: Matrix Iterative Analysis. Prentice Hall, Englewood Cliffs (1962)
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© 2007 Springer-Verlag Berlin Heidelberg
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Schwandt, H. (2007). Two-Stage Interval Krawczyk-Schwarz Methods with Applications to Nonlinear Parabolic PDE. In: Gervasi, O., Gavrilova, M.L. (eds) Computational Science and Its Applications – ICCSA 2007. ICCSA 2007. Lecture Notes in Computer Science, vol 4707. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74484-9_25
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DOI: https://doi.org/10.1007/978-3-540-74484-9_25
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-74482-5
Online ISBN: 978-3-540-74484-9
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