Mclean, William; Thomée, Vidar 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 ANZIAM J. 53, No. 2, 134-155 (2011). 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. Reviewer: Petr Tichý (Prague) Cited in 2 Documents 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 Keywords:heat equation; Laplace transform; finite elements; quadrature; Richardson iteration; conjugate gradient method; preconditioning; semidiscretization; algorithm Citations:Zbl 1068.65146; Zbl 1195.65122; Zbl 1416.65381 Software:PyAMG PDFBibTeX XMLCite \textit{W. Mclean} and \textit{V. Thomée}, ANZIAM J. 53, No. 2, 134--155 (2011; Zbl 1264.65046) Full Text: DOI arXiv 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 [3] DOI: 10.1137/090776081 · Zbl 1233.65067 · doi:10.1137/090776081 [4] DOI: 10.1002/nla.499 · Zbl 1199.65112 · doi:10.1002/nla.499 [5] DOI: 10.1023/A:1026000105893 · Zbl 1033.65015 · doi:10.1023/A:1026000105893 [6] DOI: 10.1137/040611045 · Zbl 1116.65063 · doi:10.1137/040611045 [7] DOI: 10.1093/imanum/23.2.269 · Zbl 1022.65108 · doi:10.1093/imanum/23.2.269 [8] DOI: 10.1007/BF01386412 · Zbl 0702.65034 · doi:10.1007/BF01386412 [9] DOI: 10.1090/S0025-5718-99-01098-4 · Zbl 0936.65109 · doi:10.1090/S0025-5718-99-01098-4 [10] DOI: 10.1145/992200.992206 · Zbl 1072.65037 · doi:10.1145/992200.992206 [11] DOI: 10.1137/0712047 · Zbl 0319.65025 · doi:10.1137/0712047 [12] DOI: 10.1093/imanum/drm039 · Zbl 1145.65022 · doi:10.1093/imanum/drm039 [13] DOI: 10.1137/S0895479800380386 · Zbl 1044.65029 · doi:10.1137/S0895479800380386 [14] DOI: 10.1216/JIE-2010-22-1-57 · Zbl 1195.65122 · doi:10.1216/JIE-2010-22-1-57 [15] DOI: 10.1093/imanum/drp004 · Zbl 1416.65381 · doi:10.1093/imanum/drp004 [16] DOI: 10.1145/200979.200986 · Zbl 0886.65023 · doi:10.1145/200979.200986 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.