On the convergence of a numerical method for a hypersingular integral equation on a closed surface

We consider a linear integral equation with a hypersingular integral treated in the sense of the Hadamard finite value. This equation arises in the solution of the Neumann boundary value problem for the Laplace equation with a representation of a solution in the form of a double-layer potential. We consider the case in which the interior or exterior boundary value problem is solved in a domain; whose boundary is a smooth closed surface, and an integral equation is written out on that surface. For the integral operator in that equation, we suggest quadrature formulas like the method of vortical frames with a regularization, which provides its approximation on the entire surface for the use of a nonstructured partition. We construct a numerical scheme for the integral equation on the basis of suggested quadrature formulas, prove an estimate for the norm of the inverse matrix of the related system of linear equations and the uniform convergence of numerical solutions to the exact solution of the hypersingular integral equation on the grid.     

Some formulations coupling finite element and integral equation methods for Helmholtz equation and electromagnetism

Summary. In this paper a study of the coupling between integral equations and finite element methods is presented for two problems of propagation in frequency domain. It is shown that these problems can be viewed as multidomain problems and treated by the mean of the Schur complement technique. The complement coming from the integral equation part is expressed with the integral operators of the scattering theory. This allows to predict the behaviour of the Schur method, either primal or dual, as far as its convergence speed is concerned. Furthermore, the difference of behaviour between electromagnetism and acoustics from this point of view is explained. Received October 21, 1993 / Revised version received February 28, 1994     

On the convergence of the Nyström method for the integral equation eigenvalue problem

Summary Approximate solutions of the linear integral equation eigenvalue problem can be obtained by the replacement of the integral by a numerical quadrature formula and then collocation to obtain a linear algebraic eigenvalue problem. This method is often called the Nyström method and a framework for its error analysis was introduced by Noble [15]. In this paper the convergence of the method is considered when the integral operator is a compact operator from a Banach spaceX intoX.     

On the convergence of the multigrid method for a hypersingular integral equation of the first kind

Summary We present a multigrid method to solve linear systems arising from Galerkin schemes for a hypersingular boundary integral equation governing three dimensional Neumann problems for the Laplacian. Our algorithm uses damped Jacobi iteration, Gauss-Seidel iteration or SOR as preand post-smoothers. If the integral equation holds on a closed, Lipschitz surface we prove convergence ofV- andW-cycles with any number of smoothing steps. If the integral equation holds on an open, Lipschitz surface (covering crack problems) we show convergence of theW-cycle. Numerical experiments are given which underline the theoretical results.     

An integral equation method for the electromagnetic scattering from cavities

Consider a time‐harmonic electromagnetic plane wave incident on a cavity in a ground plane. The physical process is modelled by Maxwell's equations. In this paper, integral representations of the solutions to the model problem in both fundamental polarizations are derived and studied. Existence and uniqueness of the solutions for the integral equations are established. The integral equations approach forms a basis for numerical solution of the model problem. In particular, for each fundamental polarization, an integral formulation with Gårding‐type estimates is derived. These formulations provide a basis for variational boundary element methods for solving the cavity problem. The Gårding‐type estimates imply convergence results for conforming boundary element methods. Copyright © 2000 John Wiley & Sons, Ltd     

On radiation conditions for rough surface scattering problems

^** Email: arens{at}numathics.com^*** Email: hohage{at}math.uni-goettingen.de It is well known that Sommerfeld's radiation condition is nota valid characterization of outgoing waves for scattering problemsat rough surfaces. Instead, a radiation condition called upwardpropagating radiation condition (UPRC) is commonly used. Recently,a different radiation condition called the pole condition hasbeen investigated for scattering problems at bounded obstacles.In this paper we show the equivalence between the UPRC and thepole condition. In doing so, we give a rigorous interpretationof a formula called the angular spectrum representation forDirichlet data in the space of bounded continuous functions.     

Modified boundary integral formulations for the Helmholtz equation

In this paper we describe some modified regularized boundary integral equations to solve the exterior boundary value problem for the Helmholtz equation with either Dirichlet or Neumann boundary conditions. We formulate combined boundary integral equations which are uniquely solvable for all wave numbers even for Lipschitz boundaries Γ=∂Ω. This approach extends and unifies existing regularized combined boundary integral formulations.     

On the integral equation method for buckling loads

An initial value method for the integral equation of the column is presented for determining the buckling load of columns. The differential equation of the column is reduced to a Fredholm integral equation. An initial value problem is derived for this integral equation, which is reduced to a set of ordinary differential equations with prescribed initial conditions in order to find the Fredholm resolvent. The singularities of the resolvent occur at the eigenvalues. Integration of the equations proceeds until the integrals become excessively large, indicating that a critical load has been reached. To check this method, numerical results are given for two examples, for which the critical load is well known. One is the Euler load of a simply supported beam, and the other case is the buckling load of a cantilever beam under its own weight. The advantage of this initial value method is that it can be applied easily to solve other nonlinear problems for which the critical loads are unknown. This approach will be illustrated in future papers.     

On the integral equation formulations of some 2D contact problems

Two integral equations, representing the mechanical response of a 2D infinite plate supported along a line and subject to a transverse concentrated force, are examined. The kernels of the integral operators are of the type (x−y)ln|x−y| and (x−y)²ln|x−y|. In spite of the fact that these are only weakly singular, the two equations are studied in a more general framework, which allows us to consider also solutions having non-integrable endpoint singularities. The existence and uniqueness of solutions of the equations are discussed and their endpoint singularities detected.Since the two equations are of interest in their own right, some properties of the associated integral operators are examined in a scale of weighted Sobolev type spaces. Then, new results on the existence and uniqueness of integrable solutions of the equations that in some sense are complementary to those previously obtained are derived.     

Pseudospectral time-domain method for pressure-release rough surface scattering using a surface transformation and image method

When numerically analyzing acoustic scattering at a pressure-release rough surface, the conventional pseudospectral time domain (PSTD) method using Fourier transform requires rigorous stability conditions in order to solve the spatial derivative in the wave equation on the irregular boundaries between the two media due to the Gibbs phenomenon and short wavelength in air. To eliminate such disadvantages, a new algorithm is proposed based on the Fourier PSTD method utilizing a surface boundary transformation and an image method. Irregular surface boundaries are flattened by transformation and then an image method is applied to the half-space domain. The efficiency and accuracy of the proposed PSTD method are better than the conventional Fourier PSTD method. Numerical results are presented for a sloped and a sinusoidal pressure-release surface.     

On the convergence of the heat balance integral method

Convergence properties are established for the piecewise linear heat balance integral solution of a benchmark moving boundary problem, thus generalising earlier results [Numer. Heat Transfer 8 (1985) 373]. A convergence rate of O(n⁻¹) is identified with minor effects at large values of the Stefan number β (slow interface movement). The correct O(n^−1/2) behaviour for incident heat flux is recovered for β → 0 (pure heat conduction) as previously found [Numer. Heat Transfer 8 (1985) 373–382]. Numerical illustrations support the theoretical findings.     

A finite element method for the solution of a potential theory integral equation

This paper discusses a finite element approximation for an integral equation of the second kind deduced from a potential theory boundary value problem in two variables. The equation is shown to admit a unique solution, to be variational and coercive in the Hilbert space of functions σ ε H^1/2(Γ), f_rd γ = 0. The Galerkin method with finite elements as trial functions is shown to lead to an optimal rate of convergence.     

The integral equation method in problems of diffraction by a finite impedance section of a medium interface

A 3D problem of reflection of a plane electromagnetic wave by a local impedance section of a wavy surface is considered. The boundary value problem for the system of Maxwell’s equations in a region with an irregular boundary is reduced to solution of systems of hypersingular integral equations. A numerical algorithm is proposed for solution of these systems. Results of numerical computations are presented.     

On the convergence of the collocation method for nonlinear boundary integral equations

Recently, Galerkin and collocation methods have been analysed for some nonlinear boundary integral equations. For the collocation method it has been assumed that the nonlinearity is asymptotically linear. In this paper we remove this restriction. We shall prove the convergence of the collocation method for nonlinear boundary integral equations, when the nonlinearity has a polynomial growth condition. In addition to this the optimal order error estimates follow in L^q(Γ)-norm.     

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.     

Ellipticity of the electric field integral equation for absorbing media and the convergence of the Rao-Wilton-Glisson method

The operator of the electric field integral equation is proved to be elliptic in the case of a flat screen and absorbing media. The method of quadratic forms is applied. As a result, the Rao-Wilton-Glisson method is shown to converge in the case of a flat screen in an absorbing medium.     

On the convergence of finite difference schemes for the heat equation with concentrated capacity

Summary. We investigate the convergence of difference schemes for the one-dimensional heat equation when the coefficient at the time derivative (heat capacity) is represents the magnitude of the heat capacity concentrated at the point . An abstract operator method is developed for analyzing this equation. Estimates for the rate of convergence in special discrete energetic Sobolev's norms, compatible with the smoothness of the solution are obtained. Received November 2, 1999 / Revised version received July 24, 2000 / Published online May 4, 2001     

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.     

On the convergence of the Galerkin method for nonsmooth solutions of integral equations

Summary It is shown that the stability region of the Galerkin method includes solutions not lying in the conventional energy space. Optimal order error estimates for these nonsmooth solutions are derived. The new result is compared with the classical statement by means of the basic potential problem.     

A fast Fourier-Galerkin method solving boundary integral equations for the Helmholtz equation with exponential convergence

Numerical Algorithms - A boundary integral equation in general form will be considered, which can be used to solve Dirichlet problems for the Helmholtz equation. The goal of this paper is to...