Energy concentration and explicit Sommerfeld radiation condition for the electromagnetic Helmholtz equation

Energy concentration and Sommerfeld condition for Helmholtz equation with variable index at infinity

We consider the Helmholtz equation with a variable index of refraction n(x), which is not necessarily constant at infinity but can have an angular dependency like n(x) → n ∞ (x/|x|) as |x| → ∞. Under some appropriate assumptions on this convergence and on n ∞ we prove that the Sommerfeld condition at infinity still holds true under the explicit form R d ∇u − in 1/2 ∞ u x |x| 2 dx |x| < +∞. It i...

A radiation condition for the 2-D Helmholtz equation in stratified media

We study the 2-D Helmholtz equation in perturbed stratified media, allowing the existence of guided waves. Our assumptions on the perturbing and source terms are not too restrictive. We prove two results. Firstly, we introduce a Sommerfeld-Rellich radiation condition and prove the uniqueness of the solution for the studied equation. Then, by careful asymptotic estimates, we prove the existence ...

The Helmholtz equation with source term: High-frequency analysis, and radiation condition at infinity

This talk presents a review of results obtained in [BCKP], [CPR], and more recently in [C]. References [BCKP] and [CPR] are joints works with J.D. Benamou, T. Katsaounis, B. Perthame, respectively B. Perthame and O. Runborg. The first two papers [BCKP] and [CPR] describe the propagation of the energy along the high-frequency Helmholtz equation with source term, using a Wigner function approach....

Numerical Solution for the Helmholtz Equation with Mixed Boundary Condition

We consider the numerical solution for the Helmholtz equation in R with mixed boundary conditions. The solvability of this mixed boundary value problem is established by the boundary integral equation method. Based on the Green formula, we express the solution in terms of the boundary data. The key to the numerical realization of this method is the computation of weakly singular integrals. Nume...

Fast Solution of the Helmholtz Equation with Radiation Condition by Imbedding