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Mathematics of Computation

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A parallel method
for time-discretization of parabolic problems
based on contour integral representation
and quadrature

Authors: Dongwoo Sheen, Ian H. Sloan and Vidar Thomée
Journal: Math. Comp. 69 (2000), 177-195
MSC (1991): Primary {65M12, 65M15, 65M99}
Published electronically: April 7, 1999
MathSciNet review: 1648403
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Abstract | References | Similar Articles | Additional Information

Abstract: We treat the time discretization of an initial-value problem for a homogeneous abstract parabolic equation by first using a representation of the solution as an integral along the boundary of a sector in the right half of the complex plane, then transforming this into a real integral on the finite interval $[0,1]$, and finally applying a standard quadrature formula to this integral. The method requires the solution of a finite set of elliptic problems with complex coefficients, which are independent and may therefore be done in parallel. The method is combined with spatial discretization by finite elements.

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  • 1. P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Academic Press, New York, 1975. MR 56:7119
  • 2. J. Douglas, Jr., J. E. Santos, D. Sheen, and L. S. Bennethum, Frequency domain treatment of one-dimensional scalar waves, Math. Mod. Meth. Appl. Sci. 3 (1993), 171-194. MR 94g:65111
  • 3. J. Douglas, Jr., J. E. Santos, and D. Sheen, Approximation of scalar waves in the space-frequency domain, Math. Mod. Meth. Appl. Sci 4 (1994), 509-531. MR 95e:65089
  • 4. S. C. Eisenstat, Jr., H. E. Elman, N. H. Schultz, and A. H. Sherman, The (new) Yale Sparse Matrix Package, Elliptic Problem Solvers II (A. L. Schoenstadt and G. Birkhoff, eds.), Academic Press, New York, 1983, pp. 45-52. MR 85g:65007
  • 5. E. Gallopoulos and Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods, SIAM J. Sci. Statist. Comput. 13 (1992), 1236-1264. MR 93d:65085
  • 6. M. Hochbruck and C. Lubich, On Krylov Subspace Approximations to the Matrix Exponential Operator, SIAM J. Numer. Anal. 34 (1997), 1911-1925. MR 98h:65018
  • 7. C.-O. Lee, J. Lee, and D. Sheen, A frequency-domain method for finite element solutions of parabolic problems, RIM-GARC Preprint 97-41, Department of Mathematics, Seoul National University, 1997.
  • 8. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. MR 85g:47061
  • 9. D. Sheen and Y. Yeom, A frequency-domain parallel method for the numerical approximation of parabolic problems, RIM-GARC Preprint 96-38, Department of Mathematics, Seoul National University, 1996.
  • 10. V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer Series in Computational Mathematics, Vol. 25, Springer-Verlag, Berlin Heidelberg New York, 1997. MR 98m:65007
  • 11. R. S. Varga, Functional Analysis and Approximation Theory in Numerical Analysis, Conference Board of the Mathematical Sciences Regional Conferences Series in Applied Mathematics, No. 3, SIAM, Philadelphia, PA, 1971. MR 46:9602

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Additional Information

Dongwoo Sheen
Affiliation: Department of Mathematics, Seoul National University, Seoul 151-742, Korea

Ian H. Sloan
Affiliation: School of Mathematics, University of New South Wales, Sydney 2052, Australia

Vidar Thomée
Affiliation: Department of Mathematics, Chalmers University of Technology, S-412 96 Göte- borg, Sweden

Received by editor(s): March 26, 1998
Published electronically: April 7, 1999
Additional Notes: This work was partially supported by the Australian Research Council and the Korea Science & Engineering Foundation through the Global Analysis Research Center at Seoul National University.
Article copyright: © Copyright 1999 American Mathematical Society

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