Iterative methods for shifted positive definite linear systems and time discretization of the heat equation
Abstract
In earlier work we have studied a method for discretization in time of a parabolic problem which consists in 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, and in this paper we study iterative methods for such systems. We first consider the basic and a preconditioned version of the Richardson algorithm, and then a conjugate gradient method as well as a preconditioned version thereof.
- Publication:
-
arXiv e-prints
- Pub Date:
- November 2011
- DOI:
- 10.48550/arXiv.1111.5105
- arXiv:
- arXiv:1111.5105
- Bibcode:
- 2011arXiv1111.5105M
- Keywords:
-
- Mathematics - Numerical Analysis;
- Mathematics - Analysis of PDEs;
- 65F10;
- 65M22;
- 65M60;
- 65R10
- E-Print:
- ANZIAM J. 53: 134--155, 2011