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Mathematics of Computation

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A new superconvergent collocation method for eigenvalue problems


Author: Rekha P. Kulkarni
Journal: Math. Comp. 75 (2006), 847-857
MSC (2000): Primary 47A10, 47A58, 47A75, 65J99, 65R20
DOI: https://doi.org/10.1090/S0025-5718-06-01871-0
Published electronically: January 3, 2006
MathSciNet review: 2196995
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Abstract: Here we propose a new method based on projections for the approximate solution of eigenvalue problems. For an integral operator with a smooth kernel, using an interpolatory projection at Gauss points onto the space of (discontinuous) piecewise polynomials of degree $ \leq r-1$, we show that the proposed method exhibits an error of the order of $ 4r$ for eigenvalue approximation and of the order of $ 3r$ for spectral subspace approximation. In the case of a simple eigenvalue, we show that by using an iteration technique, an eigenvector approximation of the order $ 4r$ can be obtained. This improves upon the order $ 2r$ for eigenvalue approximation in the collocation/iterated collocation method and the orders $ r$ and $ 2r$ for spectral subspace approximation in the collocation method and the iterated collocation method, respectively. We illustrate this improvement in the order of convergence by numerical examples.


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Additional Information

Rekha P. Kulkarni
Affiliation: Department of Mathematics, Indian Institute of Technology, Powai, Mumbai 400 076, India
Email: rpk@math.iitb.ac.in

DOI: https://doi.org/10.1090/S0025-5718-06-01871-0
Keywords: Eigenvalue, spectral subspace, integral equations, collocation, Gauss points
Received by editor(s): March 2, 2003
Received by editor(s) in revised form: October 28, 2004
Published electronically: January 3, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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