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ITERATIVE SOLUTION OF SHIFTED POSITIVE-DEFINITE LINEAR SYSTEMS ARISING IN A NUMERICAL METHOD FOR THE HEAT EQUATION BASED ON LAPLACE TRANSFORMATION AND QUADRATURE

Published online by Cambridge University Press:  15 August 2012

WILLIAM MCLEAN*
Affiliation:
School of Mathematics and Statistics, The University of New South Wales, Sydney 2052, Australia (email: w.mclean@unsw.edu.au)
VIDAR THOMÉE
Affiliation:
Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, S-412 96 Gothenburg, Sweden (email: thomee@chalmers.se)
*
For correspondence; e-mail: w.mclean@unsw.edu.au
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Abstract

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In earlier work we have studied a method for discretization in time of a parabolic problem, which consists of representing the exact solution as an integral in the complex plane and then applying a quadrature formula to this integral. In application to a spatially semidiscrete finite-element version of the parabolic problem, at each quadrature point one then needs to solve a linear algebraic system having a positive-definite matrix with a complex shift. We study iterative methods for such systems, considering the basic and preconditioned versions of first the Richardson algorithm and then a conjugate gradient method.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2012

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