On the definition of nonlinear stability for numerical methods

Abstract

In this paper a new definition of nonlinear stability for the general nonlinear problem F(u)=0 and the corresponding family of discretized problems Fh(uh)=0 is given. The notion of nonlinear stability introduced by Keller and later by Lopéz-Marcos and Sanz-Serna have the disadvantage that the Lipschitz constant of the derivative of Fh(uh) has to be known which, in many applications, is not practicable. The modification here proposed allows us to use linearized stability in a ball containing the solution uh to get nonlinear stability. The usual result remains true: nonlinear stability together with consistency implies convergence.