ANZIAM J.
46 (2004), 203224

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



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
, where
is the mesh of the partition and
denotes the step of the iteration. This order
of convergence is to be compared with the orders
in the Galerkin and projection methods and
in the Sloan method. The error in eigenvector
iterates is shown to be of the order of
in the new method,
in the Galerkin and projection methods and
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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