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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

Part of: Numerical linear algebra Integral equations, integral transforms Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  15 August 2012

WILLIAM MCLEAN  and
VIDAR THOMÉE
WILLIAM MCLEAN*
Affiliation:
School of Mathematics and Statistics, The University of New South Wales, Sydney 2052, Australia (email: [email protected])
VIDAR THOMÉE
Affiliation:
Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, S-412 96 Gothenburg, Sweden (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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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.

Keywords

Laplace transformfinite elementsquadratureRichardson iterationconjugate gradient methodpreconditioning

MSC classification

Secondary: 65F10: Iterative methods for linear systems 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65R10: Integral transforms
Type
Research Article
Information
The ANZIAM Journal , Volume 53 , Issue 2 , October 2011 , pp. 134 - 155
Copyright
Copyright © Australian Mathematical Society 2012

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