Stability of high-order perturbative methods for the computation of Dirichlet-Neumann operators

In this paper we present results on the stability of perturbation methods f or the evaluation of Dirichlet-Neumann operators (DNO) defined on domains t hat are viewed as complex deformations of a basic, simple geometry. In such cases, geometric perturbation methods, based on variations of the spatial domains of definition, have long been recognized to constitute efficient an d accurate means for the approximation of DNO and, in fact, several numeric al implementations have been previously proposed. Inspired by our recent an alytical work, here we demonstrate that the convergence of these algorithms is, quite generally, limited by numerical instability. Indeed, we show tha t these standard perturbative methods for the calculation of DNO suffer fro m significant ill-conditioning which is manifest even for very smooth bound aries, and whose severity increases with boundary roughness. Moreover, and again motivated by our previous work, we introduce an alternative perturbat ive approach that we show to be numerically stable. This approach can be in terpreted as a reformulation of classical perturbative algorithms (ir suita ble independent variables), and thus it allows for both direct comparison a nd the possibility of analytic continuation of the perturbation series. It can also be related to classical (preconditioned) spectral approaches and, as such, it retains, in finite arithmetic, the spectral convergence propert ies of classical perturbative methods, albeit at a higher computational cos t las it does not take advantage of possible dimensional reductions). Still , as we show, an alternative approach such as the one we propose may be man dated in cases where substantial information is contained in high-order har monics and/or perturbation coefficients of the solution. (C) 2001 Academic Press.