A second order backward difference method with variable steps for a parabolic problem

BIT Numerical Mathematics

Volumn 38, Issue 4, 1998, Pages 644-662

  Becker, J ^a  

^a KARLSTAD UNIVERSITY   (Sweden)

Author keywords

Backward difference method; Parabolic; Stability; Variable step size

Indexed keywords

EID: 0000313480     PISSN: 00063835     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF02510406     Document Type: Article

References (22)

1. J. Becker, Studies on the Two-Step Backward Difference Method for Parabolic Problems, PhD thesis, Department of Mathematics, Chalmers University of Technology and Göteborg University, 1995.
2. J. H. Bramble, J. E. Pasciak, P. H. Sammon, and V. Thomée, Incomplete iterations in multistep backward difference methods for parabolic problems with smooth and nonsmooth data, Math. Comp. 52 (1989), pp. 339-367.
3. P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, New York, 1978.
4. M. Crouzeix, On multistep approximation of semigroups in Banach spaces, J. Comput. Appl. Math. 20 (1987), pp. 25-35.
5. M. Crouzeix and A. L. Mignot, Analyse Numérique des Équations Différentielles, Masson, 1984.
6. M. Crouzeix and P. A. Raviart, Approximation d'équations d'évolution linéaires par des mèthodes multipas, in Étude numérique des grands systèmes (Proc. Sympos., Novosibirsk, 1976), Méthodes Math, de l'Informatique, 7, Dunod, Paris, 1978, pp. 133-150.
7. G. Dahlquist, W. Liniger, and O. Nevanlinna, Stability of two-step methods for variable integration steps, SIAM J. Numer. Anal. 20 (1983), pp. 1071-1085.
8. C. Gonzáles and C. Palencia, Stability of time-stepping methods for abstract time-dependent parabolic problems, SIAM J. Numer. Anal. 35 (1998), pp. 973-989.
9. R. D. Grigorieff, Stability of multistep-methods on variable grids, Numer. Math. 42 (1983), pp. 359-377.
10. R. D. Grigorieff, Time discretization of semigroups by the variable two step BDF method, Numerical Treatment of Differential Equations, K. Strehmel, ed., Teubner, Stuttgart, 1991.
11. M. Huang and V. Thomée, Some convergence estimates for semidiscrete type schemes for time-dependent nonselfadjoint parabolic equations, Math. Comp. 37 (1981), pp. 327-346.
12. M. Huang and V. Thomée, On the backward Euler method for parabolic equations with rough initial data, SIAM J. Numer. Anal. 19 (1982), pp. 599-603.
13. M.-N. Le Roux, Semidiscretization in time for parabolic problems, Math. Comp. 33 (1979), pp. 919-931.
14. M.-N. Le Roux, Variable stepsize multistep methods for parabolic problems, SIAM J. Numer. Anal. 19 (1982), pp. 725-741.
15. W. Mclean, and V. Thomée, Numerical solution of an evolution equation with a positive type memory term, J. Austral. Math. Soc. Ser. B. 35 (1993), pp. 23-70.
16. O. Nevanlinna, Some remarks on variable step integration, Z. Angew. Math. Mech. 64 (1984), pp. 315-316.
17. C. Palencia, On the stability of variable stepsize rational approximations of holomorphic semigroups, Math. Comp. 62 (1994), pp. 93-103.
18. C. Palencia, Stability of rational multistep approximations of holomorphic semigroups, Math. Comp. 64 (1995), pp. 591-599.
19. C. Palencia and B. García-Archilla, Stability of multistep methods for sectorial operators in Banach spaces, Appl. Numer. Math. 12 (1993), pp. 503-520.
20. G. Savaré, A(Θ)-stable approximations of abstract Cauchy problems, Numer. Math. 65 (1993), pp. 319-335.
21. V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer, New York, 1997.
22. M. Zlámal, Finite element multistep discretizations of parabolic boundary value problems, Math. Comp. 29 (1975), pp. 350-359.