This paper gives a brief survey of recent developments on mathematical modeling and analysis of the open cavity scattering problems, which arise in diverse scientific areas and have significant industrial and military applications. The scattering problems are studied for the two-dimensional Helmholtz equation corresponding to the transverse magnetic or electric polarization, and the three-dimensional time-harmonic and time-domain Maxwell equations. Since these problems are imposed in open domains, a key step of the analysis is to develop transparent boundary conditions and reformulate them equivalently into boundary value problems in bounded domains. The well-posedness of weak solutions is shown for the associated variational problems by using either the Lax-Milgram theorem or the Fredholm alternative.

In a composite medium that contains close-to-touching inclusions, the pointwise values of the gradient of the voltage potential may blow up as the distance $δ$ between some inclusions tends to 0 and as the conductivity contrast degenerates. In a recent paper [9], we showed that the blow-up rate of the gradient is related to how the eigenvalues of the associated Neumann-Poincaré operator converge to $±\frac{1}{2}$ as $δ$ → 0, and on the regularity of the contact. Here, we consider two connected 2-D inclusions, at a distance $δ$ › 0 from each other. When $δ$ = 0, the contact between the inclusions is of order $m ≥ 2$. We numerically determine the asymptotic behavior of the first eigenvalue of the Neumann-Poincaré operator, in terms of $δ$ and $m$, and we check that we recover the estimates obtained in [10].

This paper concerns the reconstruction of a penetrable obstacle embedded in a waveguide using the scattered data due to point sources, which is formulated as an optimization problem. We propose a fast reconstruction method based on a carefully designed finite element scheme for the direct scattering problem. The method has several merits: 1) the linear sampling method is used to quickly obtain a good initial guess; 2) finite Fourier series are used to approximate the boundary of the obstacle, which is decoupled from the boundary used by the finite element method; and 3) the mesh is fixed and hence the stiffness matrix, mass matrix, and right hand side are assembled once and only minor changes are made at each iteration. The effectiveness of the proposed method is demonstrated by numerical examples.

In this paper, we are concerned with the coupling of finite element methods and boundary integral equation methods solving the classical fluid-solid interaction problem in two dimensions. The original transmission problem is reduced to an equivalent nonlocal boundary value problem via introducing a Dirichlet-to-Neumann mapping by the direct boundary integral equation method. We show the existence and uniqueness of the solution for the corresponding variational equation. Numerical results based on the finite element method coupled with the standard Galerkin boundary element method, the fast multipole method and the Nyström method for approximating the DtN mapping are provided to illustrate the efficiency and accuracy of the numerical schemes.

We investigate time domain boundary element methods for the wave equation in $\mathbb{R}^3$, with a view towards sound emission problems in computational acoustics. The Neumann problem is reduced to a time dependent integral equation for the hypersingular operator, and we present a priori and a posteriori error estimates for conforming Galerkin approximations in the more general case of a screen. Numerical experiments validate the convergence of our boundary element scheme and compare it with the numerical approximations obtained from an integral equation of the second kind. Computations in a half-space illustrate the influence of the reflection properties of a flat street.

The present article is concerned with the numerical solution of boundary integral equations by an adaptive wavelet boundary element method. This method approximates the solution with a computational complexity that is proportional to the solution's best $N$-term approximation. The focus of this article is on algorithmic issues which includes the crucial building blocks and details about the efficient implementation. By numerical examples for the Laplace equation and the Helmholtz equation, solved for different geometries and right-hand sides, we validate the feasibility and efficiency of the adaptive wavelet boundary element method.

This paper is concerned with the problem of scattering of time-harmonic electromagnetic waves from penetrable diffraction gratings in the 2D polarization case. We propose a new, weakly singular, integral equation formulation for the scattering problem which is proved to be uniquely solvable. A main feature of the new integral equation formulation is that it avoids the computation of the normal derivative of double-layer potentials which is difficult and time consuming. A fast numerical algorithm is also developed for the scattering problem, based on the Nyström method for the new integral equation. Numerical examples are also shown to illustrate the applicability of the new integral equation formulation.

We present a fast Galerkin spectral method to solve logarithmic singular equations on segments. The proposed method uses weighted first-kind Chebyshev polynomials. Convergence rates of several orders are obtained for fractional Sobolev spaces $\tilde{H}^{-1 ⁄ 2}$ (or $H^{-1 ⁄ 2}_{00}$). Main tools are the approximation properties of the discretization basis, the construction of a suitable Hilbert scale for weighted $L^2$-spaces and local regularity estimates. Numerical experiments are provided to validate our claims.