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

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.

[1]  M. Slodicka.  Smoothing effect and discretization in time to semilinear parabolic equations with nonsmooth data , 1991 .

[2]  E. Boschi.  Recensioni: J. L. Lions - Quelques méthodes de résolution des problémes aux limites non linéaires. Dunod, Gauthier-Vi;;ars, Paris, 1969; , 1971 .

[3]  Alexander Ostermann,et al.  Non-smooth data error estimates for linearly implicit Runge-Kutta methods , 2000 .

[4]  Alexander Ostermann,et al.  Runge-Kutta time discretization of reaction-diffusion and Navier-Stokes equations: nonsmooth-data error estimates and applications to long-time behaviour , 1996 .

[5]  Donald Estep,et al.  The discontinuous Galerkin method for semilinear parabolic problems , 1993 .

[6]  Etienne Emmrich,et al.  Error of the two-step BDF for the incompressible Navier-Stokes problem , 2004 .

[7]  Vidar Thomée,et al.  On the Discretization in Time of Semilinear Parabolic Equations with Nonsmooth Initial Data , 1987 .

[8]  M. Slodicka.  Semigroup formulation of Rothe's method: application to parabolic problems , 1992 .

[9]  C. Palencia,et al.  Stability of linear multistep methods for sectorial operators in Banach spaces , 1993 .

[10]  J. Lions.  Quelques méthodes de résolution de problèmes aux limites non linéaires , 1969 .

[11]  E. Zeidler.  Nonlinear functional analysis and its applications , 1988 .

[12]  Etienne Emmrich,et al.  Stability and Convergence of the Two-step BDF for the Incompressible Navier-Stokes Problem , 2004 .

[13]  Mechthild Thalhammer,et al.  Backward Euler discretization of fully nonlinear parabolic problems , 2002, Math. Comput..

[14]  R. Temam.  Behaviour at Time t=0 of the Solutions of Semi-Linear Evolution Equations. , 1982 .

[15]  Miloš Zlámal,et al.  Finite element methods for nonlinear parabolic equations , 1977 .

[16]  C. Lubich,et al.  Linearly implicit time discretization of non-linear parabolic equations , 1995 .

[17]  Joakim Becker,et al.  A second order backward difference method with variable steps for a parabolic problem , 1998 .

[18]  Rolf Dieter Grigorieff,et al.  Stability of multistep-methods on variable grids , 1983 .