Symmetric boundary integral formulations for Helmholtz transmission problems

In this work we propose and analyze numerical methods for the approximation of the solution of Helmholtz transmission problems in two or three dimensions. This kind of problems arises in many applications related to scattering of acoustic, thermal and electromagnetic waves. Formulations based on boundary integral methods are powerful tools to deal with transmission problems in unbounded media. Different formulations using boundary integral equations can be found in the literature. We propose here new symmetric formulations based on a paper by Martin Costabel and Ernst P. Stephan (1985), that uses the Calderón projector for the interior and exterior problems to develop closed expressions for the interior and exterior Dirichlet-to-Neumann operators. These operators are then matched to obtain an integral system that is equivalent to the Helmholtz transmission problem and uses Cauchy data on the transmission boundary as unknowns. We show how to simplify the aspect and analysis of the method by employing an additional mortar unknown with respect to the ones used in the original paper, writing it in an appropriate way to devise Krylov type iterations based on the separate Dirichlet-to-Neumann operators.     

Modified equations of the first kind for the Helmholtz equation

Integral equations of the first kind for exterior problems arising in the study of the three‐dimensional Helmholtz equation are considered. These equations are derived by seeking solutions in the form of layer potentials with modified fundamental solutions. For each first kind equation, existence and uniqueness of solution are proved with the aid of composition relations involving associated modified boundary integral operators. For the Dirichlet problem, an optimal choice of the modification coefficients is considered in order to minimize the condition number of the resulting integral operator. Copyright © 2014 John Wiley & Sons, Ltd.     

A fast numerical method for a natural boundary integral equation for the Helmholtz equation

A Neumann boundary value problem of the Helmholtz equation in the exterior circular domain is reduced into an equivalent natural boundary integral equation. Using our trigonometric wavelets and the Galerkin method, the obtained stiffness matrix is symmetrical and circulant, which lead us to a fast numerical method based on fast Fourier transform. Furthermore, we do not need to compute the entries of the stiffness matrix. Especially, our method is also efficient when the wave number k in the Helmholtz equation is very large.     

Approximation of potential integral by radial bases for solutions of Helmholtz equation

For a Helmholtz equation Δu(x) + κ ² u(x) = f(x) in a region of R ^s , s ≥ 2, where Δ is the Laplace operator and κ = a + ib is a complex number with b ≥ 0, a particular solution is given by a potential integral. In this paper the potential integral is approximated by using radial bases with the order of approximation derived.      

The method of boundary integral equations for a mixed scattering problem

In this paper we consider the scattering of an electromagnetic time-harmonic plane wave by an infinite cylinder having an open arc and a bounded domain in R² as cross section. To this end, we solve a scattering problem for the Helmholtz equation in R² where the scattering object is a combination of a crack Γ and a bounded obstacle D, and we have Dirichlet-impedance type boundary condition on Γ and Dirichlet boundary condition on ∂D (∂D∈C²). Applying potential theory, the problem can be reformulated as a boundary integral system. We establish the existence and uniqueness of a solution to the system by using the Fredholm theory.     

A boundary integral equation method for the transmission eigenvalue problem

We propose a new integral equation formulation to characterize and compute transmission eigenvalues for constant refractive index that play an important role in inverse scattering problems for penetrable media. As opposed to the recently developed approach by Cossonnière and Haddar [1,2] which relies on a two by two system of boundary integral equations our analysis is based on only one integral equation in terms of Dirichlet-to-Neumann or Robin-to-Dirichlet operators which results in a noticeable reduction of computational costs. We establish Fredholm properties of the integral operators and their analytic dependence on the wave number. Further we employ the numerical algorithm for analytic non-linear eigenvalue problems that was recently proposed by Beyn [3] for the numerical computation of transmission eigenvalues via this new integral equation.     

A boundary integral equation procedure for the mixed boundary value problem of the vector helmholtz equation

A boundary integral method is developed for the mixed boundary value problem for the vector Helmholtz equation in R³. The obtained boundary integral equations for the unknown Cauchy data build a strong elliptic system of pseudodifferential equations which can therefore be used for numerical computations using Galerkin's procedure. We show existence, uniqueness and regularity of the solution of the integral equations. Especially we give the local "edge" behavior of the solution near the submanifold which divides the Dirichlet boundary from the Neumann boundary     

Modified regularization method for the Cauchy problem of the Helmholtz equation

In this paper, the Cauchy problem for the Helmholtz equation is investigated. It is known that such problem is severely ill-posed. We propose a modified regularization method to solve it based on the solution given by the method of separation of variables. Convergence estimates are presented under two different a-priori bounded assumptions for the exact solution. Finally, numerical examples are given to show the effectiveness of the proposed numerical method.     

On the coupling of BEM and FEM for exterior problems for the Helmholtz equation

This paper deals with the coupled procedure of the boundary element method (BEM) and the finite element method (FEM) for the exterior boundary value problems for the Helmholtz equation. A circle is selected as the common boundary on which the integral equation is set up with Fourier expansion. As a result, the exterior problems are transformed into nonlocal boundary value problems in a bounded domain which is treated with FEM, and the normal derivative of the unknown function at the common boundary does not appear. The solvability of the variational equation and the error estimate are also discussed.

     

The method of integral equation for scattering problem with a mixed crack

We consider the scattering of an electromagnetic time‐harmonic plane wave by an infinite cylinder having a mixed open crack (or arc) in R² as the cross section. The crack is made up of two parts, and one of the two parts is (possibly) coated by a material with surface impedance λ. We transform the scattering problem into a system of boundary integral equations by adopting a potential approach, and establish the existence and uniqueness of a weak solution to the system by the Fredholm theory. Copyright © 2011 John Wiley & Sons, Ltd.     

On the numerical solution of a boundary integral equation for the exterior Neumann problem on domains with corners

The authors propose a “modified” Nyström method to approximate the solution of a boundary integral equation connected with the exterior Neumann problem for Laplace's equation on planar domains with corners. They prove the convergence and the stability of the method and show some numerical tests.     

Modified Tikhonov regularization method for the Cauchy problem of the Helmholtz equation

In this paper, the Cauchy problem for the Helmholtz equation is investigated. By Green’s formulation, the problem can be transformed into a moment problem. Then we propose a modified Tikhonov regularization algorithm for obtaining an approximate solution to the Neumann data on the unspecified boundary. Error estimation and convergence analysis have been given. Finally, we present numerical results for several examples and show the effectiveness of the proposed method.     

A boundary integral equation for the transmission eigenvalue problem for Maxwell equation

We propose a new integral equation formulation to characterize and compute transmission eigenvalues in electromagnetic scattering. As opposed to the approach that was recently developed by Cakoni, Haddar and Meng (2015) which relies on a two‐by‐two system of boundary integral equations, our analysis is based on only one integral equation in terms of the electric‐to‐magnetic boundary trace operator that results in a simplification of the theory and in a considerable reduction of computational costs. We establish Fredholm properties of the integral operators and their analytic dependence on the wave number. Further, we use the numerical algorithm for analytic nonlinear eigenvalue problems that was recently proposed by Beyn (2012) for the numerical computation of the transmission eigenvalues via this new integral equation.     

Quasi‐local multitrace boundary integral formulations

In the context of time harmonic wave scattering by piecewise homogeneous penetrable objects, we present a new variant of the multitrace boundary integral formulation, that differs from the local multitrace approach of (Jerez‐Hanckes and Hiptmair, 2012) [13] by the presence of regularization terms involving boundary integral operators and localized at junctions i.e. points where at least three subdomains abut. We prove well‐posedness and quasioptimal convergence of conforming Galerkin discretizations for this new formulation, and present numerical results. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 2043–2062, 2015     

The Helmholtz equation in a locally perturbed half-plane with passive boundary

^** Email: mduran{at}ing.puc.cl^*** Email: ignacio.muga{at}ucv.cl^**** Email: nedelec{at}cmapx.polytechnique.fr In this article, we study the existence and uniqueness of outgoingsolutions for the Helmholtz equation in locally perturbed half-planeswith passive boundary. We establish an explicit outgoing radiationcondition which is somewhat different from the usual Sommerfeld'sone due to the appearance of surface waves. We work with thehelp of Fourier analysis and a half-plane Green's function framework.This is an extended and detailed version of the previous articleDurán et al. (2005, The Helmholtz equation with impedancein a half-plane. C. R. Acad. Sci. Paris, Ser. I, 340, 483–488).     

Cloaking via impedance boundary condition for the 2-D Helmholtz equation

We consider control problems for the 2-D Helmholtz equation in an unbounded domain with partially coated boundary. Dirichlet boundary condition is given on one part of the boundary and the impedance boundary condition is imposed on another its part. The role of control in control problem under study is played by boundary impedance. Quadratic tracking–type functionals for the field play the role of cost functionals. Solvability of control problems is proved. The uniqueness and stability of optimal solutions with respect to certain perturbations of both cost functional and incident field are established.     

Unique continuation on a line for the Helmholtz equation

In this article, local unique continuation on a line for solutions of the Helmholtz equation is discussed. The fundamental solution of the exterior problem for the Helmholtz equation have a logarithmic singularity which behaves similar to those of the interior problem for the Laplace equation in two dimension. A Hölder-type conditional stability estimate of the proposed exterior problem for the Helmholtz equation is obtained by adopting the complex extension method in Cheng and Yamamoto [J. Cheng and M. Yamamoto, Unique continuation on a line for harmonic functions, Inverse Probl. 14 (1998), pp. 869–882]. Finally, a regularization scheme based on the collocation method is compatible with the Hölder-type stability estimate provided that the line does not intersect the boundary of the domain for both the Laplace and the Helmholtz equations.     

Estimating quadrature errors for an efficient method for quasi-singular boundary integral

The numerical resolution of the boundary integral equations applied to the differential equations of Laplace, Helmholtz and Maxwell requires the handling of quasi-singular integrals with different order of singularity. The numerical approximation of the integral equations of different kinds is made by boundary finite elements. In this paper, we present a complete survey for estimating quadrature errors for the numerical techniques proposed by Huang and Cruse [Q. Huang, T.A. Cruse, Some notes on singular integral techniques in boundary element analysis, Int. J. Numer. Methods Eng. 36 (15) (1993) 2643-2659], to calculate the quasi-singular integrals. To validate the accuracy and efficiency of these techniques and approve our study some numerical examples are presented and discussed.