Stability of linear multistep methods and applications to nonlinear parabolic problems.
Stability and convergence properties of linear multistep methods for nonlinear parabolic problems are investigated. This analysis admits variable stepsizes and is based on an abstract theory of sectorial operators and analytic semigroups in Banach spaces. This approach allows the stability factor to depend on the stepsize sequence and it must be guaranteed that this factor remains bounded by a reasonable constant. Firstly this approach was used for ordinary differential equations and further for linear parabolic problems. However, these resluts are not sufficient to obtain the convergence for nonlinear problems. So the aim of this paper is to derive the stability estimates and to develop a convergence theory of multistep methods for nonlinear problems. First stability resluts for arbitrary sectorial operators on compact time interval \( [0,T] \) are proved and then this theory is expanded to time dependent problems of type: \[ u^\prime(t) = A(t) u(t), 0<t\leq T, u(0) \text{ given }, \] with Hölder continuous operator \(A\). The obtained results are applied to semilinear parabolic problems that conclude for example reaction-diffusion or incompressible Navier-Stokes equations. Convergence results for finite times are shown. The last section is devoted to convergence resluts for fully nonlinear parabolic problems: \[ u^\prime(t) = F(t,u(t)), t>0 , u(0) \text{ given }. \] The long term behaviour of multistep discretization nearby an asymptotically stable equilibirum is studied and convergence results for smooth solutions on compact time intervals are proved.
scientific article; zbMATH DE number 2056712