Numerical analysis of an exponentially ill-conditioned boundary value problem with applications to metastable problems

Metastable behaviour, which refers to an asymptotically exponentially slow time dependent motion to the limiting steady-state solution, is often assoc iated with certain exponentially ill-conditioned singularly perturbed probl ems. As a result of this severe ill-conditioning, little is known concernin g the convergence and stability of the numerical schemes that compute metas table behaviour. In this paper, a rigorous uniform convergence analysis is given for several finite difference schemes applied to a boundary layer res onance problem, which is the simplest linear exponentially ill-conditioned boundary value problem (BVP). It is found that the numerical computation of this problem does not cause any more difficulties than other standard sing ular perturbation problems, provided that we can use sufficiently high prec ision arithmetic. The qualitative results from the detailed study of this s pecific problem are shown numerically also to be valid for other exponentia lly ill-conditioned BVPs and their corresponding time-dependent equations.