The Finite Element Tearing and Interconnecting method for the Helmholtz equation is a recent non-overlapping domain decomposition method for solving linear systems arising from the nite element discretization of Helmholtz problems in bounded domains. This method was validated on two-dimensional external problems with rst order absorbing boundary conditions. The purpose of this paper is to study...

For many wave propagation problems, it is often necessary to solve the variable coeecient Helmholtz equation in a very large domain. The one-way re-formulation based on the Dirichlet-to-Neumann (DtN) map can be used to nd the solutions of the Helmholtz equation over a large range distance. It is particularly useful for waveguide problems, since it leads to a numerical algorithm that marches in ...

with the same efficiency, but it turned out that this is a very difficult task. Textbooks mention that there are substantial difficulties, see [3, page 72], [11, page 212], [12, page 400], and also the review [7] for why in general iterative methods have difficulties when applied to the Helmholtz equation (1). Motivated by the early proposition in [2] to use the Laplacian to precondition the He...

The possibility is discussed to improve the ray approximation up to an exact representation of a wave field by the Feynman–Kac probabilistic formula (this formula gives an exact solution of the Helmholtz equation in the form of the expectation of a certain functional on the space of Brownian random walks). Some examples illustrate an application of the solutions obtained to diffraction problems...

Nonpolynomial finite element methods for Helmholtz problems have seen much attention in recent years in the engineering and mathematics community. The idea is to use instead of standard polynomials Trefftz-type basis functions that already satisfy the Helmholtz equation, such as plane waves [17], Fourier-Bessel functions [8] or fundamental solutions [4]. To approximate the inter-element interfa...

We study, in the context of the full wave equation, an approximate cloaking scheme that was previously considered for the Helmholtz equation [R. V. Kohn, D. Onofrei, M. S. Vogelius, and M. I. Weinstein, Comm. Pure Appl. Math., 63 (2010), pp. 973–1016, H.-M. Nguyen and M. S. Vogelius, Arch. Ration. Mech. Anal., 203 (2011), pp. 769–807]. This cloaking scheme consists in a combination of an absorb...

Using the characterization of last multipliers as solutions of the Liouville’s transport equation, new results are given in this approach of ODE by providing several new characterizations, e.g. in terms of Witten and Marsden differentials or adjoint vector field. Applications to Hamiltonian vector fields on Poisson manifolds and vector fields on Riemannian manifolds are presented. In Poisson ca...

Operator symbols play a pivotal role in both the exact, well-posed, one-way reformulation of solving the (elliptic) Helmholtz equation and the construction of the generalized Bremmer coupling series. The inverse square-root and square-root Helmholtz operator symbols are the initial quantities of interest in both formulations, in addition to providing the theoretical framework for the developmen...

Starting from a bound state (positive or sign-changing) solution to −∆ωm = |ωm|p−1ωm − ωm in R, ωm ∈ H(R) and solutions to the Helmholtz equation ∆u0 + λu0 = 0 in R, λ > 0, we build new Dancer’s type entire solutions to the nonlinear scalar equation −∆u = |u|p−1u − u in R.