Generalised Dirichelt-to-Neumann map in time dependent domains

We study the heat, linear Schrodinger and linear KdV equations in the domain l(t) < x < ∞, 0 < t < T, with prescribed initial and boundary conditions and with l(t) a given differentiable function. For the first two equations, we show that the unknown Neumann or Dirichlet boundary value can be computed as the solution of a linear Volterra integral equation with an explicit weakly singular kernel. This integral equation can be derived from the formal Fourier integral representation of the solution. For the linear KdV equation we show that the two unknown boundary values can be computed as the solution of a system of linear Volterra integral equations with explicit weakly singular kernels. The derivation in this case makes crucial use of analyticity and certain invariance properties in the complex spectral plane. The above Volterra equations are shown to admit a unique solution.

A fast two-grid and finite section method for a class of integral equations on the real line with application to an acoustic scattering problem in the half-plane

We consider the numerical treatment of second kind integral equations on the real line of the form ∅(s) = ∫_(-∞)^(+∞)▒〖κ(s-t)z(t)ϕ(t)dt,s=R〗 (abbreviated ϕ= ψ+K_z ϕ) in which K ϵ L_1 (R), z ϵ L_∞ (R) and ψ ϵ BC(R), the space of bounded continuous functions on R, are assumed known and ϕ ϵ BC(R) is to be determined. We first derive sharp error estimates for the finite section approximation (reducing the range of integration to [-A, A]) via bounds on (1-K_z )^(-1)as an operator on spaces of weighted continuous functions. Numerical solution by a simple discrete collocation method on a uniform grid on R is then analysed: in the case when z is compactly supported this leads to a coefficient matrix which allows a rapid matrix-vector multiply via the FFT. To utilise this possibility we propose a modified two-grid iteration, a feature of which is that the coarse grid matrix is approximated by a banded matrix, and analyse convergence and computational cost. In cases where z is not compactly supported a combined finite section and two-grid algorithm can be applied and we extend the analysis to this case. As an application we consider acoustic scattering in the half-plane with a Robin or impedance boundary condition which we formulate as a boundary integral equation of the class studied. Our final result is that if z (related to the boundary impedance in the application) takes values in an appropriate compact subset Q of the complex plane, then the difference between ϕ(s)and its finite section approximation computed numerically using the iterative scheme proposed is ≤C_1 [kh log⁡〖(1⁄kh)+(1-Θ)^((-1)⁄2) (kA)^((-1)⁄2) 〗 ] in the interval [-ΘA,ΘA](Θ<1) for kh sufficiently small, where k is the wavenumber and h the grid spacing. Moreover this numerical approximation can be computed in ≤C_2 N log⁡N operations, where N = 2A/h is the number of degrees of freedom. The values of the constants C1 and C2 depend only on the set Q and not on the wavenumber k or the support of z.

Solvability and spectral properties of integral equations on the real line: I. Weighted spaces of continuous functions

We consider in this paper the solvability of linear integral equations on the real line, in operator form (λ−K)φ=ψ, where and K is an integral operator. We impose conditions on the kernel, k, of K which ensure that K is bounded as an operator on . Let Xa denote the weighted space as |s|→∞}. Our first result is that if, additionally, |k(s,t)|⩽κ(s−t), with and κ(s)=O(|s|−b) as |s|→∞, for some b>1, then the spectrum of K is the same on Xa as on X, for 01. As an example where kernels of this latter form occur we discuss a boundary integral equation formulation of an impedance boundary value problem for the Helmholtz equation in a half-plane.

A Nyström method for a class of integral equations on the real line with applications to scattering by diffraction gratings and rough surfaces

We propose a Nystr¨om/product integration method for a class of second kind integral equations on the real line which arise in problems of two-dimensional scalar and elastic wave scattering by unbounded surfaces. Stability and convergence of the method is established with convergence rates dependent on the smoothness of components of the kernel. The method is applied to the problem of acoustic scattering by a sound soft one-dimensional surface which is the graph of a function f, and superalgebraic convergence is established in the case when f is infinitely smooth. Numerical results are presented illustrating this behavior for the case when f is periodic (the diffraction grating case). The Nystr¨om method for this problem is stable and convergent uniformly with respect to the period of the grating, in contrast to standard integral equation methods for diffraction gratings which fail at a countable set of grating periods.

On the solvability of second kind integral equations on the real line

This paper considers general second kind integral equations of the form(in operator form φ − kφ = ψ), where the functions k and ψ are assumed known, with ψ ∈ Y, the space of bounded continuous functions on R, and k such that the mapping s → k(s, · ), from R to L1(R), is bounded and continuous. The function φ ∈ Y is the solution to be determined. Conditions on a set W ⊂ BC(R, L1(R)) are obtained such that a generalised Fredholm alternative holds: If W satisfies these conditions and I − k is injective for all k ∈ W then I − k is also surjective for all k ∈ W and, moreover, the inverse operators (I − k) − 1 on Y are uniformly bounded for k ∈ W. The approximation of the kernel in the integral equation by a sequence (kn) converging in a weak sense to k is also considered and results on stability and convergence are obtained. These general theorems are used to establish results for two special classes of kernels: k(s, t) = κ(s − t)z(t) and k(s, t) = κ(s − t)λ(s − t, t), where κ ∈ L1(R), z ∈ L∞(R), and λ ∈ BC((R\{0}) × R). Kernels of both classes arise in problems of time harmonic wave scattering by unbounded surfaces. The general integral equation results are here applied to prove the existence of a solution for a boundary integral equation formulation of scattering by an infinite rough surface and to consider the stability and convergence of approximation of the rough surface problem by a sequence of diffraction grating problems of increasingly large period.

On the solvability of a class of second kind integral equations on unbounded domains

We consider integral equations of the form ψ(x) = φ(x) + ∫Ωk(x, y)z(y)ψ(y) dy(in operator form ψ = φ + Kzψ), where Ω is some subset ofRn(n ≥ 1). The functionsk,z, and φ are assumed known, withz ∈ L∞(Ω) and φ ∈ Y, the space of bounded continuous functions on Ω. The function ψ ∈ Yis to be determined. The class of domains Ω and kernelskconsidered includes the case Ω = Rnandk(x, y) = κ(x − y) with κ ∈ L1(Rn), in which case, ifzis the characteristic function of some setG, the integral equation is one of Wiener–Hopf type. The main theorems, proved using arguments derived from collectively compact operator theory, are conditions on a setW ⊂ L∞(Ω) which ensure that ifI − Kzis injective for allz ∈ WthenI − Kzis also surjective and, moreover, the inverse operators (I − Kz)−1onYare bounded uniformly inz. These general theorems are used to recover classical results on Wiener–Hopf integral operators of21and19, and generalisations of these results, and are applied to analyse the Lippmann–Schwinger integral equation.

The impedance boundary value problem for the Helmholtz equation in a half-plane

We prove unique existence of solution for the impedance (or third) boundary value problem for the Helmholtz equation in a half-plane with arbitrary L∞ boundary data. This problem is of interest as a model of outdoor sound propagation over inhomogeneous flat terrain and as a model of rough surface scattering. To formulate the problem and prove uniqueness of solution we introduce a novel radiation condition, a generalization of that used in plane wave scattering by one-dimensional diffraction gratings. To prove existence of solution and a limiting absorption principle we first reformulate the problem as an equivalent second kind boundary integral equation to which we apply a form of Fredholm alternative, utilizing recent results on the solvability of integral equations on the real line in [5].

Scattering by rough surfaces: the Dirichlet problem for the Helmholtz equation in a non-locally perturbed half-plane

We consider the Dirichlet boundary value problem for the Helmholtz equation in a non-locally perturbed half-plane, this problem arising in electromagnetic scattering by one-dimensional rough, perfectly conducting surfaces. We propose a new boundary integral equation formulation for this problem, utilizing the Green's function for an impedance half-plane in place of the standard fundamental solution. We show, at least for surfaces not differing too much from the flat boundary, that the integral equation is uniquely solvable in the space of bounded and continuous functions, and hence that, for a variety of incident fields including an incident plane wave, the boundary value problem for the scattered field has a unique solution satisfying the limiting absorption principle. Finally, a result of continuous dependence of the solution on the boundary shape is obtained.

Asymptotic behavior at infinity of solutions of multidimensional second kind integral equations

We consider second kind integral equations of the form x(s) - (abbreviated x - K x = y ), in which Ω is some unbounded subset of Rn. Let Xp denote the weighted space of functions x continuous on Ω and satisfying x (s) = O(|s|-p ),s → ∞We show that if the kernel k(s,t) decays like |s — t|-q as |s — t| → ∞ for some sufficiently large q (and some other mild conditions on k are satisfied), then K ∈ B(XP) (the set of bounded linear operators on Xp), for 0 ≤ p ≤ q. If also (I - K)-1 ∈ B(X0) then (I - K)-1 ∈ B(XP) for 0 < p < q, and (I- K)-1∈ B(Xq) if further conditions on k hold. Thus, if k(s, t) = O(|s — t|-q). |s — t| → ∞, and y(s)=O(|s|-p), s → ∞, the asymptotic behaviour of the solution x may be estimated as x (s) = O(|s|-r), |s| → ∞, r := min(p, q). The case when k(s,t) = к(s — t), so that the equation is of Wiener-Hopf type, receives especial attention. Conditions, in terms of the symbol of I — K, for I — K to be invertible or Fredholm on Xp are established for certain cases (Ω a half-space or cone). A boundary integral equation, which models three-dimensional acoustic propaga-tion above flat ground, absorbing apart from an infinite rigid strip, illustrates the practical application and sharpness of the above results. This integral equation mod-els, in particular, road traffic noise propagation along an infinite road surface sur-rounded by absorbing ground. We prove that the sound propagating along the rigid road surface eventually decays with distance at the same rate as sound propagating above the absorbing ground.

Approximate solution of second kind integral equations on infinite cylindrical surfaces

The paper considers second kind integral equations of the form $\phi (x) = g(x) + \int_S {k(x,y)} \phi (y)ds(y)$ (abbreviated $\phi = g + K\phi $), in which S is an infinite cylindrical surface of arbitrary smooth cross section. The “truncated equation” (abbreviated $\phi _a = E_a g + K_a \phi _a $), obtained by replacing S by $S_a $, a closed bounded surface of class $C^2 $, the boundary of a section of the interior of S of length $2a$, is also discussed. Conditions on k are obtained (in particular, implying that K commutes with the operation of translation in the direction of the cylinder axis) which ensure that $I - K$ is invertible, that $I - K_a $ is invertible and $(I - K_a )^{ - 1} $ is uniformly bounded for all sufficiently large a, and that $\phi _a $ converges to $\phi $ in an appropriate sense as $a \to \infty $. Uniform stability and convergence results for a piecewise constant boundary element collocation method for the truncated equations are also obtained. A boundary integral equation, which models three-dimensional acoustic scattering from an infinite rigid cylinder, illustrates the application of the above results to prove existence of solution (of the integral equation and the corresponding boundary value problem) and convergence of a particular collocation method.

On asymptotic behavior at infinity and the finite section method for integral equations on the half-line

e consider integral equations on the half-line of the form and the finite section approximation to x obtained by replacing the infinite limit of integration by the finite limit β. We establish conditions under which, if the finite section method is stable for the original integral equation (i.e. exists and is uniformly bounded in the space of bounded continuous functions for all sufficiently large β), then it is stable also for a perturbed equation in which the kernel k is replaced by k + h. The class of perturbations allowed includes all compact and some non-compact perturbations of the integral operator. Using this result we study the stability and convergence of the finite section method in the space of continuous functions x for which ()()()=−∫∞dttxt,sk)s(x0()syβxβx()sxsp+1 is bounded. With the additional assumption that ()(tskt,sk−≤ where ()()(),qsomefor,sassOskandRLkq11>+∞→=∈− we show that the finite-section method is stable in the weighted space for ,qp≤≤0 provided it is stable on the space of bounded continuous functions. With these results we establish error bounds in weighted spaces for x - xβ and precise information on the asymptotic behaviour at infinity of x. We consider in particular the case when the integral operator is a perturbation of a Wiener-Hopf operator and illustrate this case with a Wiener-Hopf integral equation arising in acoustics.

Some uniform stability and convergence results for integral equations on the real line and projection methods for their solution

The paper considers second kind equations of the form (abbreviated x=y + K2x) in which and the factor z is bounded but otherwise arbitrary so that equations of Wiener-Hopf type are included as a special case. Conditions on a set are obtained such that a generalized Fredholm alternative is valid: if W satisfies these conditions and I − Kz, is injective for each z ε W then I − Kz is invertible for each z ε W and the operators (I − Kz)−1 are uniformly bounded. As a special case some classical results relating to Wiener-Hopf operators are reproduced. A finite section version of the above equation (with the range of integration reduced to [−a, a]) is considered, as are projection and iterated projection methods for its solution. The operators (where denotes the finite section version of Kz) are shown uniformly bounded (in z and a) for all a sufficiently large. Uniform stability and convergence results, for the projection and iterated projection methods, are obtained. The argument generalizes an idea in collectively compact operator theory. Some new results in this theory are obtained and applied to the analysis of projection methods for the above equation when z is compactly supported and k(s − t) replaced by the general kernel k(s,t). A boundary integral equation of the above type, which models outdoor sound propagation over inhomogeneous level terrain, illustrates the application of the theoretical results developed.

Sparse space-time boundary element methods for the heat equation

The goal of this work is the efficient solution of the heat equation with Dirichlet or Neumann boundary conditions using the Boundary Elements Method (BEM). Efficiently solving the heat equation is useful, as it is a simple model problem for other types of parabolic problems. In complicated spatial domains as often found in engineering, BEM can be beneficial since only the boundary of the domain has to be discretised. This makes BEM easier than domain methods such as finite elements and finite differences, conventionally combined with time-stepping schemes to solve this problem. The contribution of this work is to further decrease the complexity of solving the heat equation, leading both to speed gains (in CPU time) as well as requiring smaller amounts of memory to solve the same problem. To do this we will combine the complexity gains of boundary reduction by integral equation formulations with a discretisation using wavelet bases. This reduces the total work to O(h

A Galerkin boundary element method for high frequency scattering by convex polygons

In this paper we consider the problem of time-harmonic acoustic scattering in two dimensions by convex polygons. Standard boundary or finite element methods for acoustic scattering problems have a computational cost that grows at least linearly as a function of the frequency of the incident wave. Here we present a novel Galerkin boundary element method, which uses an approximation space consisting of the products of plane waves with piecewise polynomials supported on a graded mesh, with smaller elements closer to the corners of the polygon. We prove that the best approximation from the approximation space requires a number of degrees of freedom to achieve a prescribed level of accuracy that grows only logarithmically as a function of the frequency. Numerical results demonstrate the same logarithmic dependence on the frequency for the Galerkin method solution. Our boundary element method is a discretization of a well-known second kind combined-layer-potential integral equation. We provide a proof that this equation and its adjoint are well-posed and equivalent to the boundary value problem in a Sobolev space setting for general Lipschitz domains.

A combined BIE-FE method for the Stokes equations

A numerical algorithm for the biharmonic equation in domains with piecewise smooth boundaries is presented. It is intended for problems describing the Stokes flow in the situations where one has corners or cusps formed by parts of the domain boundary and, due to the nature of the boundary conditions on these parts of the boundary, these regions have a global effect on the shape of the whole domain and hence have to be resolved with sufficient accuracy. The algorithm combines the boundary integral equation method for the main part of the flow domain and the finite-element method which is used to resolve the corner/cusp regions. Two parts of the solution are matched along a numerical ‘internal interface’ or, as a variant, two interfaces, and they are determined simultaneously by inverting a combined matrix in the course of iterations. The algorithm is illustrated by considering the flow configuration of ‘curtain coating’, a flow where a sheet of liquid impinges onto a moving solid substrate, which is particularly sensitive to what happens in the corner region formed, physically, by the free surface and the solid boundary. The ‘moving contact line problem’ is addressed in the framework of an earlier developed interface formation model which treats the dynamic contact angle as part of the solution, as opposed to it being a prescribed function of the contact line speed, as in the so-called ‘slip models’. Keywords: Dynamic contact angle; finite elements; free surface flows; hybrid numerical technique; Stokes equations.