Iterative solution of shifted positive-definite linear systems arising in a numerical method for the heat equation based on Laplace transformation and quadrature

William McLean; Vidar Thomee

doi:10.21914/anziamj.v53i0.5248

Authors

William McLean The University of New South Wales Sydney 2050
Vidar Thomee Chalmers University of Technology and University of Gothenburg

DOI:

https://doi.org/10.21914/anziamj.v53i0.5248

Keywords:

Laplace transform, finite elements, quadrature, Richardson iteration, conjugate gradient method, preconditioning.

Abstract

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. doi:10.1017/S1446181112000107

Author Biographies

William McLean, The University of New South Wales Sydney 2050

Senior Lecturer, School of Mathematics and Statistics

Vidar Thomee, Chalmers University of Technology and University of Gothenburg

Emeritus Professor, Mathematical Sciences