Stability and error of the variable two-step BDF for semilinear parabolic problems

Journal of Applied Mathematics and Computing - Tập 19 - Trang 33-55 - 2005
Etienne Emmrich1
1Institut für Mathematik, Technische Universität Berlin, Berlin, Germany

Tóm tắt

The temporal discretisation of a moderate semilinear parabolic problem in an abstract setting by the two-step backward differentiation formula with variable step sizes is analysed. Stability as well as optimal smooth data error estimates are derived if the ratios of adjacent step sizes are bounded from above by 1.91.

Tài liệu tham khảo

J. Becker,A second order backward difference method with variable steps for a parabolic problem, BIT,38 (1998) 4, pp. 644–662. M. Crouzeix and V. Thomée,On the discretization in time of semilinear parabolic equations with nonsmooth initial data, Math. Comp.,49 (1987) 180, pp. 359–377. E. Emmrich,Stability and convergence of the two-step BDF for the incompressible Navier-Stokes problem, Int. J. Nonlinear Sci. Numer. Simul.,5 (2004) 3, pp. 199–209. E. Emmrich,Error of the two-step BDF for the incompressible Navier-Stokes problem, M2AN, Math. Model. Numer. Anal.38 (2004) 5, 757–764. D. Estep and S. Larsson,The discontinuous Galerkin method for semilinear parabolic problems, M2AN, Math Model. Numer. Anal.,27 (1993) 1, 35–54. H. Fujita and T. Suzuki,Evolution Problems, in Handbook of Numerical Analysis, Vol. II: Finite Element Methods (Part 1), P. G. Ciarlet and J.-L. Lions, eds. Elsevier, Amsterdam, 1991, pp. 789–928. C. González, A. Ostermann, C. Palencia, and M. Thalhammer,Backward Euler discretization of fully nonlinear parabolic problems. Math. Comput.,71 (2002) 237, pp. 125–145. R. D. Grigorieff,Stability of multistep-methods on variable grids, Numer. Math.,42 (1983), pp. 359–377. R. D. Grigorieff,Time discretization of semigroups by the variable two-step BDF method, in Numerical treatment of differential equations, Sel. Papers NUMDIFF-5, Halle, 1989, B. G. Teubner, Leipzig, 1991, pp. 204–216. R. D. Grigorieff,On the variable grid two-step BDF method for parabolic equations, Preprint 426, Fachber. Mathematik, Techn. Univ. Berlin, 1995. M.-N. Le Roux,Variable step size multistep methods for parabolic problems, SIAM J. Numer. Anal.,19 (1982) 4, pp. 725–741. J.-L. Lions,Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod Gauthier-Villars, Paris, 1969. C. Lubich and A. Ostermann,Linearly implicit time discretization of non-linear parabolic equations, IMA J. Numer. Anal.,15 (1995) 4, pp. 555–583. C. Lubich and A. Ostermann,Runge-Kutta time discretization of reaction-diffusion and Navier-Stokes equations: nonsmooth-data error estimates and applications to long-time behaviour, Appl. Numer. Math.,22 (1996), pp. 279–292. R. Temam,Behaviour at time t=0 of the solutions of semi-linear evolution equations, J. Diff. Eqs.,43 (1982), pp. 73–92. A. Ostermann and M. Thalhammer,Non-smooth data error estimates for linearly implicit Runge-Kutta methods, IMA J. Numer. Anal.,20 (2000) 20, pp. 167–184. C. Palencia and B. García-Archilla,Stability of linear multistep methods for sectorial operators in Banach spaces, Appl. Numer. Math.,12 (1993) 6, pp. 503–520. M. Slodička,Smoothing effect and discretization in time to semilinear parabolic equations with nonsmooth data, Comment. Math. Univ. Carol.,32 (1991) 4, 703–713. M. Slodička,Semigroup formulation of Rothe's method: application to parabolic problems, Comment. Math. Univ. Carol.,33 (1992) 2, 245–260. V. Thomée,Galerkin Finite Element Methods for Parabolic Problems, Springer, Berlin, 1997. E. Zeidler,Nonlinear Functional Analysis and its Applications I, Springer, New York, 1992. M. Zlámal,Finite element methods for nonlinear parabolic equations, RAIRO Analyse Numérique,11 (1977) 1, pp. 93–107.