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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. (English) Zbl 1264.65046

Authors’ abstract: In earlier work [IMA J. Numer. Anal. 24, No. 3, 439–463 (2004; Zbl 1068.65146); ibid. 30, No. 1, 208–230 (2010; Zbl 1416.65381); J. Integral Equations Appl. 22, No. 1, 57–94 (2010; Zbl 1195.65122)], 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.

MSC:

65F10 Iterative numerical methods for linear systems
65M22 Numerical solution of discretized equations for initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65R10 Numerical methods for integral transforms
44A10 Laplace transform
35A22 Transform methods (e.g., integral transforms) applied to PDEs
65F08 Preconditioners for iterative methods
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs

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References:

[1] DOI: 10.1007/s00211-005-0657-7 · Zbl 1097.65131 · doi:10.1007/s00211-005-0657-7
[2] DOI: 10.1093/imanum/24.3.439 · Zbl 1068.65146 · doi:10.1093/imanum/24.3.439
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