A Rigorous Numerical Analysis of the Transformed Field Expansion Method

Boundary perturbation methods, in which the deviation of the problem geometry from a simple one is taken as the small quantity, have received considerable attention in recent years due to an enhanced understanding of their convergence properties. One approach to deriving numerical methods based upon these ideas leads to Bruno and Reitich's generalization [Proc. Roy. Soc. Edinburgh Sect. A, 122 (1992), pp. 317-340] of Rayleigh and Rice's classical algorithm giving the “method of variation of boundaries” which is very fast and accurate within its domain of applicability. Treating problems outside this domain (e.g., boundary perturbations which are large and/or rough) led Nicholls and Reitich [Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), pp. 1411-1433] to design the “transformed field expansions” (TFE) method, and the rigorous numerical analysis of these recursions is the subject of the current work. This analysis is based upon analyticity estimates for the TFE expansions coupled to the convergence of Fourier-Legendre Galerkin methods. This powerful and flexible analysis is extended to a wide range of problems including those governed by Laplace and Helmholtz equations, and the equations of traveling free-surface ideal fluid flow.