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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

Part of: Integral equations, integral transforms Partial differential equations, initial value and time-dependent initial-boundary value problems Numerical linear algebra

Published online by Cambridge University Press:  15 August 2012

WILLIAM MCLEAN  and
VIDAR THOMÉE
WILLIAM MCLEAN*
Affiliation:
School of Mathematics and Statistics, The University of New South Wales, Sydney 2052, Australia (email: w.mclean@unsw.edu.au)
VIDAR THOMÉE
Affiliation:
Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, S-412 96 Gothenburg, Sweden (email: thomee@chalmers.se)
*
✉For correspondence; e-mail: w.mclean@unsw.edu.au
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Abstract

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In earlier work 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.

Keywords

Laplace transformfinite elementsquadratureRichardson iterationconjugate gradient methodpreconditioning

MSC classification

Secondary: 65F10: Iterative methods for linear systems 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65R10: Integral transforms

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 53 , Issue 2 , October 2011 , pp. 134 - 155
Copyright
Copyright © Australian Mathematical Society 2012

References

[1]Bell, W. N., Olson, L. N. and Schroder, J. B., “PyAMG: algebraic multigrid solvers in Python v2.0”, http://www.pyamg.org.Google Scholar
[2]Benzi, M. and Bertaccini, D., “Block preconditioning of real-valued iterative algorithms for complex linear systems”, IMA J. Numer. Anal. 28 (2008) 598–618; doi:10.1093/imanum/drm039.CrossRefGoogle Scholar
[3]Davis, T. A., “Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method”, ACM Trans. Math. Software 30 (2004) 196–199; doi:10.1145/992200.992206.CrossRefGoogle Scholar
[4]Freund, R., “On conjugate gradient type methods and polynomial preconditioners for a class of complex non-Hermitian matrices”, Numer. Math. 57 (1990) 285–312; doi:10.1007/BF01386412.CrossRefGoogle Scholar
[5]Gavrilyuk, I. P. and Makarov, V. L., “Exponentially convergent algorithms for the operator exponential with applications to inhomogeneous problems in Banach spaces”, SIAM J. Numer. Anal. 43 (2005) 2144–2171; doi:10.1137/040611045.CrossRefGoogle Scholar
[6]Geuzaine, C. and Remacle, J.-F., “Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities”, http://www.geuz.org/gmsh.Google Scholar
[7]in ’t Hout, K. J. and Weideman, J. A. C., “A contour integral method for the Black–Scholes and Heston equations”, SIAM J. Sci. Comput. 33 (2011) 763–785; doi:10.1137/090776081.CrossRefGoogle Scholar
[8]Jones, M. T. and Plassmann, P. E., “Algorithm 740: Fortran subroutines to compute improved incomplete Cholesky factorizations”, ACM Trans. Math. Software 21 (1995) 18–19; doi:10.1145/200979.200986.CrossRefGoogle Scholar
[9]McLean, W., Sloan, I. H. and Thomée, V., “Time discretization via Laplace tranformation of an integro-differential equation of parabolic type”, Numer. Math. 102 (2006) 497–522; doi:10.1007/s00211-005-0657-7.CrossRefGoogle Scholar
[10]McLean, W. and Thomée, V., “Time discretization of an evolution equation via Laplace transforms”, IMA J. Numer. Anal. 24 (2004) 439–463; doi:10.1093/imanum/24.3.439.CrossRefGoogle Scholar
[11]McLean, W. and Thomée, V., “Maximum-norm error analysis of a numerical solution via Laplace transformation and quadrature of a fractional order evolution equation”, IMA J. Numer. Anal. 30 (2010) 208–230; doi:10.1093/imanum/drp004.CrossRefGoogle Scholar
[12]McLean, W. and Thomée, V., “Numerical solution via Laplace transforms of a fractional-order evolution equation”, J. Integral Equations Appl. 22 (2010) 57–94; doi:10.1216/JIE-2010-22-1-57.CrossRefGoogle Scholar
[13]McLean, W. and Thomée, V., “Iterative methods for shifted positive definite linear systems and time discretization of the heat equation”, Preprint, 2011, http://arxiv.org/abs/1111.5105.Google Scholar
[14]Meerbergen, K., “The solution of parametrized symmetric linear systems”, SIAM J. Matrix Anal. Appl. 24 (2003) 1038–1059; doi:10.1137/S0895479800380386.CrossRefGoogle Scholar
[15]Paige, C. C. and Saunders, M. A., “Solution of sparse indefinite systems of linear equations”, SIAM J. Numer. Anal. 12 (1975) 617–629; doi:10.1137/0712047.CrossRefGoogle Scholar
[16]Sheen, D., Sloan, I. H. and Thomée, V., “A parallel method for time-discretization of parabolic problems based on contour integral representation and quadrature”, Math. Comp. 69 (2000) 177–195; doi:10.1090/S0025-5718-99-01098-4.CrossRefGoogle Scholar
[17]Sheen, D., Sloan, I. H. and Thomée, V., “A parallel method for time-discretization of parabolic equations based on Laplace transformation and quadrature”, IMA J. Numer. Anal. 23 (2003) 269–299; doi:10.1093/imanum/23.2.269.CrossRefGoogle Scholar
[18]Simoncini, V., “Restarted full orthogonalization method for shifted linear systems”, BIT Numer. Math. 43 (2003) 459–466; doi:10.1023/A:1026000105893.CrossRefGoogle Scholar
[19]Simoncini, V. and Szyld, D. B., “Recent computational developments in Krylov subspace methods for linear systems”, Numer. Linear Algebra Appl. 14 (2007) 1–59; doi:10.1002/nla.499.CrossRefGoogle Scholar