The convergence of the cascadic conjugate-gradient method applied to elliptic problems in domains with re-entrant corners
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- by Vladimir Shaidurov and Lutz Tobiska PDF
- Math. Comp. 69 (2000), 501-520 Request permission
Abstract:
We study the convergence properties of the cascadic conjugate-gradient method (CCG-method), which can be considered as a multilevel method without coarse-grid correction. Nevertheless, the CCG-method converges with a rate that is independent of the number of unknowns and the number of grid levels. We prove this property for two-dimensional elliptic second-order Dirichlet problems in a polygonal domain with an interior angle greater than $\pi$. For piecewise linear finite elements we construct special nested triangulations that satisfy the conditions of a “triangulation of type $(h,\gamma ,L)$” in the sense of I. Babuška, R. B. Kellogg and J. Pitkäranta. In this way we can guarantee both the same order of accuracy in the energy norm of the discrete solution and the same convergence rate of the CCG-method as in the case of quasiuniform triangulations of a convex polygonal domain.References
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Additional Information
- Vladimir Shaidurov
- Affiliation: Institute of Computational Modelling, Siberian Branch of the Russian Academy of Sciences, Krasnoyarsk 660036, Russia
- Email: shidurov@cckr.krasnoyarsk.su
- Lutz Tobiska
- Affiliation: Otto-von-Guericke-Universität Magdeburg, Postfach 4120, D-39016 Magdeburg, Germany
- Email: tobiska@mathematik.uni-magdeburg.de
- Received by editor(s): November 11, 1997
- Received by editor(s) in revised form: July 10, 1998
- Published electronically: March 18, 1999
- Additional Notes: The research was supported by the Deutsche Forschungsgemeinschaft.
- © Copyright 2000 American Mathematical Society
- Journal: Math. Comp. 69 (2000), 501-520
- MSC (1991): Primary 65F10; Secondary 65N30
- DOI: https://doi.org/10.1090/S0025-5718-99-01138-2
- MathSciNet review: 1653982