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

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      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.

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      Journal
      1111.5105
      10.1017/S1446181112000107

      Analysis, Numerical & Computational mathematics

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