Incomplete iterations in multistep backward difference methods for parabolic problems with smooth and nonsmooth data

Bramble, James H.; Pasciak, Joseph E.; Sammon, Peter H.; Thomée, Vidar

doi:10.1090/S0025-5718-1989-0962207-8

Incomplete iterations in multistep backward difference methods for parabolic problems with smooth and nonsmooth data
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by James H. Bramble, Joseph E. Pasciak, Peter H. Sammon and Vidar Thomée PDF

Math. Comp. 52 (1989), 339-367 Request permission

Abstract:

Backward difference methods for the discretization of parabolic boundary value problems are considered in this paper. In particular, we analyze the case when the backward difference equations are only solved ’approximately’ by a preconditioned iteration. We provide an analysis which shows that these methods remain stable and accurate if a suitable number of iterations (often independent of the spatial discretization and time step size) are used. Results are provided for the smooth as well as non-smooth initial data cases. Finally, the results of numerical experiments illustrating the algorithms’ performance on model problems are given.

References

Garth A. Baker, James H. Bramble, and Vidar Thomée, Single step Galerkin approximations for parabolic problems, Math. Comp. 31 (1977), no. 140, 818–847. MR 448947, DOI 10.1090/S0025-5718-1977-0448947-X
James H. Bramble, Discrete methods for parabolic equations with time-dependent coefficients, Numerical methods for partial differential equations (Proc. Adv. Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1978) Publ. Math. Res. Center Univ. Wisconsin, vol. 42, Academic Press, New York-London, 1979, pp. 41–52. MR 558215

Comput. Methods Appl. Mech. Engrg.

James H. Bramble and Joseph E. Pasciak, New convergence estimates for multigrid algorithms, Math. Comp. 49 (1987), no. 180, 311–329. MR 906174, DOI 10.1090/S0025-5718-1987-0906174-X
J. H. Bramble, J. E. Pasciak, and A. H. Schatz, An iterative method for elliptic problems on regions partitioned into substructures, Math. Comp. 46 (1986), no. 174, 361–369. MR 829613, DOI 10.1090/S0025-5718-1986-0829613-0
J. H. Bramble, J. E. Pasciak, and A. H. Schatz, The construction of preconditioners for elliptic problems by substructuring. I, Math. Comp. 47 (1986), no. 175, 103–134. MR 842125, DOI 10.1090/S0025-5718-1986-0842125-3
J. H. Bramble, J. E. Pasciak, and A. H. Schatz, The construction of preconditioners for elliptic problems by substructuring. II, Math. Comp. 49 (1987), no. 179, 1–16. MR 890250, DOI 10.1090/S0025-5718-1987-0890250-4
James H. Bramble, Joseph E. Pasciak, and Alfred H. Schatz, The construction of preconditioners for elliptic problems by substructuring. III, Math. Comp. 51 (1988), no. 184, 415–430. MR 935071, DOI 10.1090/S0025-5718-1988-0935071-X
James H. Bramble, Joseph E. Pasciak, and Alfred H. Schatz, The construction of preconditioners for elliptic problems by substructuring. IV, Math. Comp. 53 (1989), no. 187, 1–24. MR 970699, DOI 10.1090/S0025-5718-1989-0970699-3
James H. Bramble and Peter H. Sammon, Efficient higher order single step methods for parabolic problems. I, Math. Comp. 35 (1980), no. 151, 655–677. MR 572848, DOI 10.1090/S0025-5718-1980-0572848-X
J. H. Bramble, A. H. Schatz, V. Thomée, and L. B. Wahlbin, Some convergence estimates for semidiscrete Galerkin type approximations for parabolic equations, SIAM J. Numer. Anal. 14 (1977), no. 2, 218–241. MR 448926, DOI 10.1137/0714015
James H. Bramble and Vidar Thomée, Discrete time Galerkin methods for a parabolic boundary value problem, Ann. Mat. Pura Appl. (4) 101 (1974), 115–152. MR 388805, DOI 10.1007/BF02417101

Conjugate Gradient Methods for Partial Differential Equations

M. Crouzeix and P.-A. Raviart, Approximation d’équations d’évolution linéaires par des méthodes multipas, Étude numérique des grands systèmes (Proc. Sympos., Novosibirsk, 1976) Méthodes Math. Inform., vol. 7, Dunod, Paris, 1978, pp. 133–150 (French). MR 517853
Colin W. Cryer, On the instability of high order backward-difference multistep methods, Nordisk Tidskr. Informationsbehandling (BIT) 12 (1972), 17–25. MR 311112, DOI 10.1007/bf01932670
Jim Douglas Jr., Todd Dupont, and Richard E. Ewing, Incomplete iteration for time-stepping a Galerkin method for a quasilinear parabolic problem, SIAM J. Numer. Anal. 16 (1979), no. 3, 503–522. MR 530483, DOI 10.1137/0716039
Todd Dupont, Richard P. Kendall, and H. H. Rachford Jr., An approximate factorization procedure for solving self-adjoint elliptic difference equations, SIAM J. Numer. Anal. 5 (1968), 559–573. MR 235748, DOI 10.1137/0705045
C. William Gear, Numerical initial value problems in ordinary differential equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. MR 0315898
F. B. Hildebrand, Introduction to numerical analysis, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1956. MR 0075670
Stephen L. Keeling, Galerkin/Runge-Kutta discretizations for parabolic equations with time-dependent coefficients, Math. Comp. 52 (1989), no. 186, 561–586. MR 958873, DOI 10.1090/S0025-5718-1989-0958873-3
Marie-Noëlle Le Roux, Semidiscretization in time for parabolic problems, Math. Comp. 33 (1979), no. 147, 919–931. MR 528047, DOI 10.1090/S0025-5718-1979-0528047-2

Problèmes aux Limites non Homogènes et Applications

Iterative Methods for the Solution of a Linear Operator Equation in Hilbert Space—A Survey

Galerkin Finite Element Methods for Parabolic Problems

Harry Yserentant, On the multilevel splitting of finite element spaces, Numer. Math. 49 (1986), no. 4, 379–412. MR 853662, DOI 10.1007/BF01389538
Miloš Zlámal, Finite element multistep discretizations of parabolic boundary value problems, Math. Comp. 29 (1975), 350–359. MR 371105, DOI 10.1090/S0025-5718-1975-0371105-2