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Efficient higher order single step methods for parabolic problems. I


Authors: James H. Bramble and Peter H. Sammon
Journal: Math. Comp. 35 (1980), 655-677
MSC: Primary 65N30; Secondary 65M15
DOI: https://doi.org/10.1090/S0025-5718-1980-0572848-X
MathSciNet review: 572848
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Abstract: Some efficient, high order methods are discussed for approximating the solution of an initial boundary value problem for a homogeneous parabolic equation with time dependent coefficients. The methods are based on Galerkin-type approximations in the spacial variables and single step methods in the time variable. The equations defining the time-stepping procedure are solved only approximately, however. A preconditioned iterative technique is used for this purpose. The resulting algorithm is shown to produce optimal order approximations using only the order of work required by the single step method applied to the parabolic problem with time independent coefficients.


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  • [1] O. AXELSSON, On Preconditioning and Convergence Acceleration in Sparse Matrix Problems, CERN (European Organization for Nuclear Research), Geneva, 1974.
  • [2] G. A. BAKER, J. H. BRAMBLE & V. THOMÉE, "Single step Galerkin approximations for parabolic problems," Math. Comp., v. 31. 1977, pp. 818-847. MR 0448947 (56:7252)
  • [3] J. DOUGLAS, JR. & T. DUPONT, "Alternating direction methods on rectangles," Numerical Solution of Partial Differential Equations-II (B. Hubbard, Ed.), Academic Press, New York, 1971. MR 0273830 (42:8706)
  • [4] J. DOUGLAS, JR., T. DUPONT & R. EWING, "Incomplete iteration for time-stepping a Galerkin method for a quasilinear parabolic problem," SIAM J. Numer. Anal., v. 16, 1979, pp. 503-522. MR 530483 (80f:65117)
  • [5] A. FRIEDMAN, Partial Differential Equations, Krieger, Huntington, New York, 1976. MR 0454266 (56:12517)
  • [6] J. L. LIONS & E. MAGENES, Nonhomogeneous Boundary Value Problems and Applications, Vol. II, Springer-Verlag, New York, 1973. MR 0350179 (50:2672)
  • [7] N. NASSIF & J. DESCLOUX, "Stability study for time-dependent linear parabolic equations and its application to Hermitian methods," Topics in Numerical Analysis III (J. Miller, Ed.), Academic Press, New York, 1977. MR 0657787 (58:31875)
  • [8] P. H. SAMMON, Convergence Estimates for Semidiscrete Parabolic Equation Approximations, Mathematics Research Center Technical Survey Report No. 2053, 1980.
  • [9] RICHARD S. VARGA, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, N.J., 1962. MR 0158502 (28:1725)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1980-0572848-X
Keywords: Parabolic equations, Galerkin methods, higher order methods
Article copyright: © Copyright 1980 American Mathematical Society

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