Orthogonal spline collocation Laplacemodified and alternatingdirection methods for parabolic problems on rectangles
Authors:
Bernard Bialecki and Ryan I. Fernandes
Journal:
Math. Comp. 60 (1993), 545573
MSC:
Primary 65N35; Secondary 65M12, 65M70, 65N12
MathSciNet review:
1176704
Fulltext PDF Free Access
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Additional Information
Abstract: A complete stability and convergence analysis is given for two and threelevel, piecewise Hermite bicubic orthogonal spline collocation, Laplacemodified and alternatingdirection schemes for the approximate solution of linear parabolic problems on rectangles. It is shown that the schemes are unconditionally stable and of optimalorder accuracy in space and time.
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B. Bialecki and X. C. Cai, error bounds for piecewise Hermite bicubic orthogonal spline collocation schemes, submitted.
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A. Celia and G.
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for parabolic equations. III. Nonrectangular domains, Numer. Methods
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space variable, Lecture Notes in Mathematics, Vol. 385,
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𝐶¹piecewisepolynomial spaces. MR 0483559
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R. I. Fernandes, Alternating direction finite element methods for solving time dependent problems, Ph. D. Thesis, University of Kentucky, Lexington, KY, 1991.
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Ryan
I. Fernandes and Graeme
Fairweather, Analysis of alternating direction collocation methods
for parabolic and hyperbolic problems in two space variables, Numer.
Methods Partial Differential Equations 9 (1993),
no. 2, 191–211. MR 1203061
(94c:65105), http://dx.doi.org/10.1002/num.1690090207
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element approximations on rectangles, Comput. Math. Appl.
6 (1980), no. 1, Special Issue, 45–50. MR 604084
(82b:65124), http://dx.doi.org/10.1016/08981221(80)900589
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, A comparison of alternatingdirection collocation methods for the transport equation, New Concepts in Finite Element Analysis (T. J. R. Hughes et al., eds.), AMSVol. 44, American Society of Mechanical Engineers, New York, 1981, pp. 169177.
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Milton
Lees, A priori estimates for the solutions of difference
approximations to parabolic partial differential equations, Duke Math.
J. 27 (1960), 297–311. MR 0121998
(22 #12725)
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Peter
Percell and Mary
Fanett Wheeler, A 𝐶¹ finite element collocation method
for elliptic equations, SIAM J. Numer. Anal. 17
(1980), no. 5, 605–622. MR 588746
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A. Samarskiĭ, Teoriya raznostnykh skhem, Izdat.
“Nauka”, Moscow, 1977 (Russian). MR 0483271
(58 #3288)
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A. A. Samarskii and A. B. Gulin, Stability of difference schemes, "Nauka", Moscow, 1973 (Russian).
 [1]
 V. K. Bangia, C. Bennett, A. Reynolds, R. Raghavan, and G. Thomas, Alternating direction collocation methods for simulating reservoir performance, Paper SPE 7414, 53rd SPE Fall Technical Conference and Exhibition, Houston, Texas, 1978.
 [2]
 B. Bialecki and X. C. Cai, error bounds for piecewise Hermite bicubic orthogonal spline collocation schemes, submitted.
 [3]
 B. Bialecki, G. Fairweather, and K. R. Bennett, Fast direct solvers for piecewise Hermite bicubic orthogonal spline collocation equations, SIAM J. Numer. Anal. 29 (1992), 156173. MR 1149090 (92j:65031)
 [4]
 B. Bialecki and R. I. Fernandes, Implementation of orthogonal spline collocation Laplacemodified and alternatingdirection methods for parabolic problems, in preparation.
 [5]
 G. Birkhoff, M. H. Schultz, and R. S. Varga, Piecewise Hermite interpolation in one and two variables with applications to partial differential equations, Numer. Math. 11 (1968), 232256. MR 0226817 (37:2404)
 [6]
 J. F. Botha and M. Celia, The alternating direction collocation approximation, Proceedings of the Eighth South African Symposium on Numerical Mathematics, Durban, South Africa, July, 1982, pp. 1326. MR 749407 (85k:65097)
 [7]
 M. A. Celia, L. R. Ahuja, and G. F. Pinder, Orthogonal collocation and alternatingdirection methods for unsaturated flow, Adv. Water Resources 10 (1987), 178187.
 [8]
 M. A. Celia and G. F. Pinder, Collocation solution of the transport equation using a locally enhanced alternating direction formulation, Unification of Finite Element Methods (H. Kardestuncer, ed.), Elsevier Science Publishers, New York, 1984, pp. 303320. MR 845620 (88c:65105)
 [9]
 , An analysis of alternatingdirection collocation methods for parabolic equations, Numer. Methods Partial Differential Equations 1 (1985), 5770. MR 868051 (88a:65109)
 [10]
 , Generalized alternatingdirection collocation methods for parabolic equations. I. Spatially varying coefficients, Numer. Methods Partial Differential Equations 6 (1990), 193214. MR 1062376 (91g:65254)
 [11]
 , Generalized alternatingdirection collocation methods for parabolic equations. II. Transport equations with application to seawater intrusion problems, Numer. Methods Partial Differential Equations 6 (1990), 215230. MR 1062377 (91g:65255)
 [12]
 , Generalized alternatingdirection collocation methods for parabolic equations. III. Nonrectangular domains, Numer. Methods Partial Differential Equations 6 (1990), 231243. MR 1062378 (91g:65256)
 [13]
 M. A. Celia, G. F. Pinder, and L. J. Hayes, Alternatingdirection collocation solution to the transport equation, Proc. Third Internat. Conf. Finite Elements in Water Resources (S. Y. Wang et al., eds.), Univ. of Mississippi, Oxford, MS, 1980, pp. 3.363.48.
 [14]
 J. H. Cerutti and S. V. Parter, Collocation methods for parabolic partial differential equations in one space dimension, Numer. Math. 26 (1976), 227254. MR 0433922 (55:6892)
 [15]
 J. E. Dendy, An analysis of some Galerkin schemes for the solution of nonlinear timedependent probledms, SIAM J. Numer. Anal. 28 (1975), 541565. MR 0418477 (54:6516)
 [16]
 J. C. Diaz, G. Fairweather, and P. Keast, FORTRAN packages for solving certain almost block diagonal linear systems by modified alternate row and column elimination, ACM Trans. Math. Software 9 (1983), 358375. MR 791972
 [17]
 , Algorithm 603COLROW and ARCECO: FORTRAN packages for solving certain almost block diagonal linear systems by modified alternate row and column elimination, ACM Trans. Math. Software 9 (1983), 376380. MR 791973
 [18]
 J. Douglas, Jr. and T. Dupont, Alternating direction Galerkin methods on rectangles, Numerical Solution of Partial Differential EquationsII (B. Hubbard, ed.), Academic Press, New York, 1971, pp. 133214. MR 0273830 (42:8706)
 [19]
 , Collocation methods for parabolic equations in a single space variable, Lecture Notes in Math., vol. 385, SpringerVerlag, New York, 1974. MR 0483559 (58:3551)
 [20]
 R. I. Fernandes, Alternating direction finite element methods for solving time dependent problems, Ph. D. Thesis, University of Kentucky, Lexington, KY, 1991.
 [21]
 R. I. Fernandes and G. Fairweather, Analysis of alternating direction collocation methods for parabolic and hyperbolic problems in two space variables, Numer. Methods Partial Differential Equations 9 (1993), (to appear). MR 1203061 (94c:65105)
 [22]
 L. J. Hayes, An alternatingdirection collocation method for finite element approximations on rectangles, Comput. Math. Appl. 6 (1980), 4550. MR 604084 (82b:65124)
 [23]
 , A comparison of alternatingdirection collocation methods for the transport equation, New Concepts in Finite Element Analysis (T. J. R. Hughes et al., eds.), AMSVol. 44, American Society of Mechanical Engineers, New York, 1981, pp. 169177.
 [24]
 M. Lees, A priori estimates for the solutions of difference approximations to parabolic partial differential equations, Duke Math. J. 27 (I960), 297311. MR 0121998 (22:12725)
 [25]
 P. Percell and M. F. Wheeler, A finite element collocation method for elliptic equations, SIAM J. Numer. Anal. 17 (1980), 605622. MR 588746 (82c:65081)
 [26]
 A. A. Samarskii, Theory of difference schemes, "Nauka", Moscow, 1977 (Russian). MR 0483271 (58:3288)
 [27]
 A. A. Samarskii and A. B. Gulin, Stability of difference schemes, "Nauka", Moscow, 1973 (Russian).
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718199311767047
PII:
S 00255718(1993)11767047
Article copyright:
© Copyright 1993 American Mathematical Society
