Spectral refinement using a new projection method

ANZIAM J. 46 (2004), 203-224

Spectral refinement using a new projection method

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

and

N. Gnaneshwar
Department of Mathematics
Indian Institute of Technology
Bombay, Powai
Mumbai 400 076
India
gnanesh@math.iitb.ac.in

Abstract

In this paper we consider two spectral refinement schemes, elementary and double iteration, for the approximation of eigenelements of a compact operator using a new approximating operator. We show that the new method performs better than the Galerkin, projection and Sloan methods. We obtain precise orders of convergence for the approximation of eigenelements of an integral operator with a smooth kernel using either the orthogonal projection onto a spline space or the interpolatory projection at Gauss points onto a discontinuous piecewise polynomial space. We show that in the double iteration scheme the error for the eigenvalue iterates using the new method is of the order of $h^{4r}(h^{3r})^k$ , where

is the mesh of the partition and $k = 0,1,2,\dots$ denotes the step of the iteration. This order of convergence is to be compared with the orders $h^{2r}(h^{r})^k$ in the Galerkin and projection methods and $h^{2r} {(h^{2r})}^k$ in the Sloan method. The error in eigenvector iterates is shown to be of the order of $h^{3r}(h^{3r})^k$ in the new method, $h^{r}(h^{r})^k$ in the Galerkin and projection methods and $h^{2r}(h^{2r})^k$ in the Sloan method. Similar improvement is observed in the case of the elementary iteration. We show that these orders of convergence are preserved in the corresponding discrete methods obtained by replacing the integration by a numerical quadrature formula. We illustrate this improvement in the order of convergence by numerical examples.

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