A-posteriori existence in adaptive computations

Abstract

This short note demonstrates that it is not necessary to assume the existence of exact solutions in an a-posteriori error analysis. If the residual of a stable numerical solution is sufficiently small there exists a nearby exact solution for which an a-posteriori error estimate holds.\ud \ud We first develop the idea in an abstract Banach space setting and then demonstrate some further practical details at the nonlinear Laplace equation.\ud \ud The author acknowledges the financial support received from the European research project HPRN-CT-2002-00284: New Materials, Adaptive Systems and their Nonlinearities. Modelling, Control and Numerical Simulation, and the kind hospitality of Carlo Lovadina (University of Pavia)