A Rigorous Numerical Analysis of the Transformed Field Expansion Method

Abstract

Abstract. 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. 1. Introduction. Perturbation techniques have been a crucial tool for scientists and engineers for hundreds of years, many classical and contemporary books, e.g., [25,4], have been devoted to their explication. One very important subclass of such methods are those designed for problems where the perturbation occurs in its geometry; of particular interest for us are partial differential equations posed on complicated domains which are "nearly" simple (e.g., rectangular, circular, and spherical). Such considerations go back at least as far as Stokes [51] in the context of free-surface ideal fluid mechanics (water waves) and Rayleigh [43] regarding the scattering of linear acoustic waves by an irregular obstacle, for example.Quite recently, this class of "boundary perturbation" (BP) methods has been carefully reexamined for use as highly accurate computational methodologies for the simulation of engineering problems. For instance, for the problem of electromagnetic or acoustic scattering by irregular obstacles, Bruno and Reitich [7,8,9,10,11,12] generalized the method of Rayleigh [43] and Rice [46] to arbitrarily high order and demonstrated that this method can deliver highly accurate answers with computational complexities which match the current state-of-the-art. These methods were largely ignored for many years due to an incomplete understanding of their convergence properties. However, these issues were resolved by Bruno and Reitich [6] (and later clarified by one of the authors and Reitich [31]) using the theory of complex

A Multi-Domain Spectral IPDG Method for Helmholtz Equation with High Wave Number

“…With a proper boundary perturbation technique (or the so-called transformed field expansion) [28], the Helmholtz equation (1.2) with exact DtN boundary condition can be reduced to a sequence of Helmholtz equations in a separable domain, e.g., an annulus and a spherical shell (cf. [14,29,30,33]). Shen and Wang [35] provided a rigorous analysis of the spectral-Galerkin method with explicit dependence of the errors on the wave number for the Helmholtz equation in an annulus or spherical shell with exact DtN boundary condition.…”

“…Shen and Wang [35] provided a rigorous analysis of the spectral-Galerkin method with explicit dependence of the errors on the wave number for the Helmholtz equation in an annulus or spherical shell with exact DtN boundary condition. The analysis for full coupled spectral-Galerkin and boundary perturbation was conducted in [30]. Indeed, within the domain of applicability of the boundary perturbation method, this approach has proven to be fast and accurate.…”

Spectral Stability of Deep Two‐Dimensional Gravity‐Capillary Water Waves

“…These {λ n , w n } have been approximated using the stable and highly (spectrally) accurate [19] method of "Transformed Field Expansions" (TFE), which was used to such great effect by one of the authors with F. Reitich [18,7] to simulate the underlying traveling waves. We refer the interested reader to [7] in particular for demonstrations of the capabilities of the TFE approach versus other Boundary Perturbation Methods including its favorable operation counts, lack of substantial numerical ill-conditioning, and applicability to large traveling wave profiles via numerical analytic continuation.…”

An Efficient and Stable Spectral-Element Method for Acoustic Scattering by an Obstacle

“…The numerical analysis of this spectral-element method (4.13) to (4.10)-(4.12) can be carried out in a similar fashion as in [22]. Then, the complete error analysis for the transformed field expansion algorithm presented in this paper follows from the general framework established in [14].…”

“…13) 14) where the exact expressions for F 1n , F 2n , Q n and J n are given in the Appendix, see (A.3)-(A.6).…”