A new finite element realization of the perfectly matched layer method for Helmholtz scattering problems on polygonal domains in two dimensions

In this paper we propose a new finite element realization of the Perfectly Matched Layer method (PML-method). Our approach allows to deal with a wide class of polygonal domains and with certain types of inhomogeneous exterior domains. Among the covered inhomogeneities are open waveguide structures playing an essential role in integrated optics. We give a detailed insight into implementation aspects. Numerical examples show exponential convergence behavior to the exact solution with the thickness of the PML sponge layer.     

Minimal area problems for functions with integral representation

We study the minimization problem for the Dirichlet integral in some standard classes of analytic functions. In particular, we solve the minimal areaa ₂-problem for convex functions and for typically real functions. The latter gives a new solution to the minimal areaa ₂-problem for the classS of normalized univalent functions in the unit disc. Supported by NSF grant DMS-0412908.     

Well‐conditioned boundary integral formulations for high‐frequency elastic scattering problems in three dimensions

We construct and analyze a family of well‐conditioned boundary integral equations for the Krylov iterative solution of three‐dimensional elastic scattering problems by a bounded rigid obstacle. We develop a new potential theory using a rewriting of the Somigliana integral representation formula. From these results, we generalize to linear elasticity the well‐known Brakhage–Werner and combined field integral equation formulations. We use a suitable approximation of the Dirichlet‐to‐Neumann map as a regularizing operator in the proposed boundary integral equations. The construction of the approximate Dirichlet‐to‐Neumann map is inspired by the on‐surface radiation conditions method. We prove that the associated integral equations are uniquely solvable and possess very interesting spectral properties. Promising analytical and numerical investigations, in terms of spherical harmonics, with the elastic sphere are provided. Copyright © 2014 John Wiley & Sons, Ltd.     

Bang-bang property for Bolza problems in two dimensions

Consider the following Bolza problem: $$\begin{gathered} \min \int {h(x,u) dt,} \hfill \\ \dot x = F(x) + uG(x), \hfill \\ \left| u \right| \leqslant 1, x \in \Omega \subset \mathbb{R}^2 , \hfill \\ x(0) = x_0 , x(1) = x_1 . \hfill \\ \end{gathered} $$ We show that, under suitable assumptions onF, G, h, all optimal trajectories are bang-bang. The proof relies on a geometrical approach that works for every smooth two-dimensional manifold. As a corollary, we obtain existence results for nonconvex optimization problems.     

A volume integral equation method for periodic scattering problems for anisotropic Maxwell's equations

This paper presents a volume integral equation method for an electromagnetic scattering problem for three-dimensional Maxwell's equations in the presence of a biperiodic, anisotropic, and possibly discontinuous dielectric scatterer. Such scattering problem can be reformulated as a strongly singular volume integral equation (i.e., integral operators that fail to be weakly singular). In this paper, we firstly prove that the strongly singular volume integral equation satisfies a Gårding-type estimate in standard Sobolev spaces. Secondly, we rigorously analyze a spectral Galerkin method for solving the scattering problem. This method relies on the periodization technique of Gennadi Vainikko that allows us to efficiently evaluate the periodized integral operators on trigonometric polynomials using the fast Fourier transform (FFT). The main advantage of the method is its simple implementation that avoids for instance the need to compute quasiperiodic Green's functions. We prove that the numerical solution of the spectral Galerkin method applied to the periodized integral equation converges quasioptimally to the solution of the scattering problem. Some numerical examples are provided for examining the performance of the method.     

A spectral integral representation for decomposable operators

For the operators commuting with a spectral measure, a Bartle type integral representation in the spirit of the one provided by the spectral theorem for normal operators is obtained. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On free boundary problems in two dimensions

Let L be a linear elliptic operator in two dimensions with analytic coefficients and of second order, andu(x, y) a solution of Lu=0 in a simply connected domain ω with rectifiable boundary Γ. Suppose ψ(x, y) analytic on ω∪Γ and L ψ≠0 there.H is shown that ifu and ψ coincide with first derivatives on an open portion Γ₀ of Γ, then Γ₀ permits the representation λ=x (θ),y=y (θ) withx(θ),y(θ)analytic functions of a real parameter θ.     

An adaptive fast direct solver for boundary integral equations in two dimensions

We describe an algorithm for the rapid direct solution of linear algebraic systems arising from the discretization of boundary integral equations of potential theory in two dimensions. The algorithm is combined with a scheme that adaptively rearranges the parameterization of the boundary in order to minimize the ranks of the off-diagonal blocks in the discretized operator, thus obviating the need for the user to supply a parameterization r of the boundary for which the distance ‖r(s)−r(t)‖ between two points on the boundary is related to their corresponding distance |s−t| in the parameter space. The algorithm has an asymptotic complexity of , where N is the number of nodes in the discretization. The performance of the algorithm is illustrated with several numerical examples.     

A heuristic for solving large bin packing problems in two and three dimensions

The more-dimensional bin packing problem (BPP) considered here requires packing a set of rectangular-shaped items into a minimum number of identical rectangular-shaped bins. All items may be rotated and the guillotine cut constraint has to be respected. A straightforward heuristic is presented that is based on a method for the container loading problem following a wall-building approach and on a method for the one-dimensional BPP. 1,800 new benchmark instances are introduced for the two-dimensional and three-dimensional BPP. The instances include more than 1,500 items on average. Applied to these very large instances, the heuristic generates solutions of acceptable quality in short computation times. Moreover, the influence of different instance parameters on the solution quality is investigated by an extended computational study.     

A parallel method for time-discretization of parabolic problems based on contour integral representation and quadrature

We treat the time discretization of an initial-value problem for a homogeneous abstract parabolic equation by first using a representation of the solution as an integral along the boundary of a sector in the right half of the complex plane, then transforming this into a real integral on the finite interval , and finally applying a standard quadrature formula to this integral. The method requires the solution of a finite set of elliptic problems with complex coefficients, which are independent and may therefore be done in parallel. The method is combined with spatial discretization by finite elements.

     

Recurrent dimensions of quasi-periodic solutions for nonlinear evolution equations

In this paper we introduce recurrent dimensions of discrete dynamical systems and we give upper and lower bounds of the recurrent dimensions of the quasi-periodic orbits. We show that these bounds have different values according to the algebraic properties of the frequency and we investigate these dimensions of quasi-periodic trajectories given by solutions of a nonlinear PDE.

     

An integral equation technique for scattering problems with mixed boundary conditions

This paper presents an integral formulation for Helmholtz problems with mixed boundary conditions. Unlike most integral equation techniques for mixed boundary value problems, the proposed method uses a global boundary charge density. As a result, Calderón identities can be utilized to avoid the use of hypersingular integral operators. Numerical results illustrate the performance of the proposed solution technique.     

A new frequency‐uniform coercive boundary integral equation for acoustic scattering

A new boundary integral operator is introduced for the solution of the soundsoft acoustic scattering problem, i.e., for the exterior problem for the Helmholtz equation with Dirichlet boundary conditions. We prove that this integral operator is coercive in L²(Γ) (where Γ is the surface of the scatterer) for all Lipschitz star‐shaped domains. Moreover, the coercivity is uniform in the wavenumber k = ω/c, where ω is the frequency and c is the speed of sound. The new boundary integral operator, which we call the “star‐combined” potential operator, is a slight modification of the standard combined potential operator, and is shown to be as easy to implement as the standard one. Additionally, to the authors' knowledge, it is the only second‐kind integral operator for which convergence of the Galerkin method in L²(Γ) is proved without smoothness assumptions on Γ except that it is Lipschitz. The coercivity of the star‐combined operator implies frequency‐explicit error bounds for the Galerkin method for any approximation space. In particular, these error estimates apply to several hybrid asymptoticnumerical methods developed recently that provide robust approximations in the high‐frequency case. The proof of coercivity of the star‐combined operator critically relies on an identity first introduced by Morawetz and Ludwig in 1968, supplemented further by more recent harmonic analysis techniques for Lipschitz domains. © 2011 Wiley Periodicals, Inc.     

A new integral equation approach to the Neumann problem in acoustic scattering

We suggest a new approach of reduction of the Neumann problem in acoustic scattering to a uniquely solvable Fredholm integral equation of the second kind with weakly singular kernel. To derive this equation we placed an additional boundary with an appropriate boundary condition inside the scatterer. The solution of the problem is obtained in the form of a single layer potential on the whole boundary. The density in the potential satisfies a uniquely solvable Fredholm integral equation of the second kind and can be computed by standard codes. Copyright © 2001 John Wiley & Sons, Ltd.