The Localized Finite Element Method and Its Application to the Two-Dimensional Sea-Keeping Problem

Abstract

International audienceWe study the various convergence properties of the localized finite element method, which is devoted to the numerical solution of linear problems set in unbounded domains. The problem under consideration is the so-called sea-keeping problem, i.e., the determination of the motion of ships undergoing the action of an incident swell, but in fact the method applies to numerous other problems. Copyright © 1988 Society for Industrial and Applied Mathematic

Computing guided modes for an unbounded stratified medium in integrated optics

“…With the same method as described in [10] we prove that, for λ < γ d 1 (k), the operator Q ± λ is continuous from…”

“…We use a series development of the solution in the exterior domain Ω e = Ω\Ω b which is an exact representation of the solution on the vertical boundaries Σ ± = {±a}× [−d, d]. It is the localized finite element method describe by Lenoir and Tounsi [10] in hydrodynamics, and then by Bonnet and Gmati [1,4,7] in guided optics when the medium is homogeneous or a diopter.…”

“…This method was introduced and studied by Lenoir and Tounsi [10] in hydrodynamics, then, in guided optics, by Bonnet [1,4] for the optical fiber with an homogeneous cladding and Gmati [7] for the diopter. For a complete stratified medium, this method leads to difficulties of the same order than the calculation of the Green function but we have adapted it in a method which is possible to use.…”

A spectral method for the Stokes problem in three-dimensional unbounded domains

“…Most of them rely on the use of integral equations and fast algorithms (see for instance [15], [4], [14], and references therein). An alternative approach is to introduce an artificial boundary and a coupling between a numerical method in the interior and either an eigenmode decomposition or an integral equation on the artificial boundary ( [8], [11]). Successful computations using spectral methods in two dimensions have been reported in [2], and theoretical formalism for the two-dimensional Stokes problem has been introduced in [9].…”

A multiscale correction method for local singular perturbations of the boundary

“…The results are much better and we used this method in the the model computations shown in next section. (iii) Numerous more accurate artificial boundary conditions are available, most of them developed in the framework of wave propagation: local transparent conditions (see [6,7]) or non-local "exactly absorbing" conditions using an integral representation (localized finite element method [11]). The functions u 0 and V 1 being computed, it remains to perform the superposition of u 0 (x) with the correcting term εχ(x)V 1 ( x ε ).…”