| Abstract: | The object of this paper is to establish the relation between stability and convergence of the numerical methods for the evolution equation ut- 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 thenprove the fact that stability and compatibility imply the uniform convergence onthe infinite time region. Thus, we establish a general frame in order to prove thelong-time convergence. This frame includes finite element methods and finite difference methods of the evolution equations, especially the semilinear parabolic andhyperbolic partial differential equations. As applications of these results we provethe estimates obtained by Larsson 5] and Sanz-serna and Stuart 6]. |
