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Algorithms without accuracy saturation for evolution equations in Hilbert and Banach spaces

Authors: Ivan P. Gavrilyuk and Volodymyr L. Makarov
Journal: Math. Comp. 74 (2005), 555-583
MSC (2000): Primary 65J10, 65M70; Secondary 35K90, 35L90
Published electronically: October 27, 2004
MathSciNet review: 2114638
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Abstract: We consider the Cauchy problem for the first and the second order differential equations in Banach and Hilbert spaces with an operator coefficient $A(t)$ depending on the parameter $t$. We develop discretization methods with high parallelism level and without accuracy saturation; i.e., the accuracy adapts automatically to the smoothness of the solution. For analytical solutions the rate of convergence is exponential. These results can be viewed as a development of parallel approximations of the operator exponential $e^{-tA}$ and of the operator cosine family $\cos{\sqrt{A} t}$ with a constant operator $A$ possessing exponential accuracy and based on the Sinc-quadrature approximations of the corresponding Dunford-Cauchy integral representations of solutions or the solution operators.

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Additional Information

Ivan P. Gavrilyuk
Affiliation: Berufsakademie Thüringen, Am Wartenberg 2, D-99817 Eisenach, Germany

Volodymyr L. Makarov
Affiliation: National Academy of Sciences of Ukraine, Institute of Mathematics, Tereschen- kivska 3, 01601 Kiev, Ukraine

Keywords: Evolution equation, parameter dependent operator, algorithms without accuracy saturation, exponentially convergent algorithms, Sinc-methods
Received by editor(s): January 21, 2003
Received by editor(s) in revised form: February 26, 2004
Published electronically: October 27, 2004
Article copyright: © Copyright 2004 American Mathematical Society