The Stability and Convergence of Computing Long-Time Behaviour

Abstract

The object of this paper is to establish the relation between stability and convergence of the numerical methods for the evolution equation $u_t-Au-f(u)=g(t)$ on Banach space $V$, and to prove the long-time error estimates for the approximation solutions. At first, we give the definition of long-time stability, and then prove the fact that stability and compatibility imply the uniform convergence on the infinite time region. Thus, we establish a general frame in order to prove the long-time convergence. This frame includes finite element methods and finite difference methods of the evolution equations, especially the semilinear parabolic and hyperbolic partial differential equations. As applications of these results we prove the estimates obtained by Larsson [5] and Sanz-serna and Stuart [6].