|
A parallel method for time-discretization of parabolic problems based on contour integral representation and quadrature
Author(s):
Dongwoo
Sheen;
Ian
H.
Sloan;
Vidar
Thomée.
Journal:
Math. Comp.
69
(2000),
177-195.
MSC (1991):
Primary {65M12, 65M15, 65M99}
Posted:
April 7, 1999
Retrieve article in:
PDF
This article is available free of charge
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 ![$[0,1]$](/mcom/2000-69-229/S0025-5718-99-01098-4/gif-abstract/img14.gif) , 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.
References:
- 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
Similar Articles:
Retrieve articles in Mathematics of Computation
with MSC
(1991):
{65M12, 65M15, 65M99}
Retrieve articles in all Journals with MSC
(1991):
{65M12, 65M15, 65M99}
Additional Information:
Dongwoo
Sheen
Affiliation:
Department of Mathematics, Seoul National University, Seoul 151-742, Korea
Email:
sheen@math.snu.ac.kr
Ian
H.
Sloan
Affiliation:
School of Mathematics, University of New South Wales, Sydney 2052, Australia
Email:
sloan@maths.unsw.edu.au
Vidar
Thomée
Affiliation:
Department of Mathematics, Chalmers University of Technology, S-412 96 Göteborg, Sweden
Email:
thomee@math.chalmers.se
DOI:
10.1090/S0025-5718-99-01098-4
PII:
S 0025-5718(99)01098-4
Received by editor(s):
March 26, 1998
Posted:
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.
Copyright of article:
Copyright
1999,
American Mathematical Society
|
Comments: webmaster@ams.org