On the Roles of "Stability" and "Convergence" in Semidiscrete Projection Methods for Initial-Value Problems

Consider the initial value problem ~u(t) = Au(t) + f(t), t > 0, (1.2) "(0) = "0, where A is a linear operator taking D(A) C X into X, where A" is a Banach space. Consider also semidiscrete numerical methods of the form: find UpA]t): [0, T] ■—► XN such that dUN (1.1') Sf = ANUN + PNl(1-2*) £/jv<0) = u% e xN, where X^¡ is a finite dimensional subspace and P¡y is a projector onto X^¡. The study of such numerical methods may be related to the approximation of semigroups and Laplace transform methods making use of the resolvent operators (A XI)~ , (A^ hltf) . The basic results require stability or weak stability and give convergence rates of the same order as in the steady state problems.