Iterative methods for shifted positive definite linear systems and time discretization of the heat equation

TL;DR

This paper investigates iterative algorithms for solving shifted positive definite linear systems arising from time discretization of the heat equation, focusing on methods like Richardson and conjugate gradient, with and without preconditioning.

Contribution

It introduces and analyzes iterative methods tailored for complex-shifted positive definite systems in heat equation discretization, enhancing computational efficiency.

Findings

Preconditioned Richardson method improves convergence.

Conjugate gradient method effectively solves shifted systems.

Preconditioning significantly accelerates iterative solutions.

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.