Regular and well-posed formulation of the boundary integral method for a singular biharmonic problem

By adequate choice of a fundamental solution, the singular point of the solution is excluded from the integral equations. The use of a special differential operator yields a well-posed formulation of the system of two integral equations. Moreover, the application of the symmetry principle for biharmonic functions improves the efficiency of the method. Finally, the results are used to compute the coefficients of the William's series (stress intensity factors) which is the eigenfunction expansion of the solution around the singular point.The research was supported in part by the Technion VPR Fund-M. R. Saulson Research Fund.     

A Nyström method for boundary integral equations in domains with corners

Summary We give a convergence and error analysis for a Nyström method on a graded mesh for the numerical solution of boundary integral equations for the harmonic Dirichlet problem in plane domains with corners.

Dedicated to Professor L. Collatz on the occassion of his 80th birthday     

Fast Fourier-Galerkin methods for solving singular boundary integral equations: Numerical integration and precondition

We develop a fast fully discrete Fourier-Galerkin method for solving a class of singular boundary integral equations. We prove that the number of multiplications used in generating the compressed matrix is O(nlog³n), and the solution of the proposed method preserves the optimal convergence order O(n^−t), where n is the order of the Fourier basis functions used in the method and t denotes the degree of regularity of the exact solution. Moreover, we propose a preconditioning which ensures the numerical stability when solving the preconditioned linear system. Numerical examples are presented to confirm the theoretical estimates and to demonstrate the approximation accuracy and computational efficiency of the proposed algorithm.     

A fast Fourier-collocation method for second boundary integral equations

In this paper we develop a fast collocation method for second boundary integral equations by the trigonometric polynomials. We propose a convenient way to compress the dense matrix representation of a compact integral operator with a smooth kernel under the Fourier basis and the corresponding collocation functionals. The compression leads to a sparse matrix with only O(nlog²n) number of nonzero entries, where 2n+1 denotes the order of the matrix. Thus we develop a fast Fourier-collocation method. We prove that the fast Fourier-collocation method gives the optimal convergence order up to a logarithmic factor. Moreover, we design a fast scheme for solving the corresponding truncated linear system. We establish that this algorithm preserves the quasi-optimal convergence of the approximate solution with requiring a number of O(nlog³n) multiplications.     

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     

An integral equation method for the electromagnetic scattering from a scatterer on an absorbing plane

Consider a time-harmonic electromagnetic plane wave incident on a scatterer on a grounded absorbing plane modelized as an infinite impedance plane. In this paper, a new integral representation formula is rigorously derived. Existence and uniqueness of weak solutions for the model problem are also established. The proof of existence is based on an extension of the Hodge decomposition technique to open boundaries. The results reported in this paper form a basis for numerical solutions of the electromagnetic scattering problem from a scatterer on an absorbing plane.     

The fast Fourier transform and the numerical solution of one-dimensional boundary integral equations

Summary Here we present a fully discretized projection method with Fourier series which is based on a modification of the fast Fourier transform. The method is applied to systems of integro-differential equations with the Cauchy kernel, boundary integral equations from the boundary element method and, more generally, to certain elliptic pseudodifferential equations on closed smooth curves. We use Gaussian quadratures on families of equidistant partitions combined with the fast Fourier transform. This yields an extremely accurate and fast numerical scheme. We present complete asymptotic error estimates including the quadrature errors. These are quasioptimal and of exponential order for analytic data. Numerical experiments for a scattering problem, the clamped plate and plane estatostatics confirm the theoretical convergence rates and show high accuracy.     

On the boundary integral equations for the crack opening displacement of flat cracks

The boundary integral equations for the crack opening displacement in acoustic and elastic scattering problems are discussed in the case of flat cracks by means of the Fourier analysis technique. The pseudo-differential nature of the hypersingular integral operators is shown and their symbols explicited. It is then proved that the variational problems assocaited with these BIE are well-posed in a Sobolev functional framework which is closely linked with the elastic energy. A decomposition of the vector integral equation in the elastic case into scalar integral equations is obtained as a by-product of the variational formulation.     

Numerical recovery of Robin boundary from boundary measurements for the Laplace equation

Based on an integral equation formulation, we present numerical methods for the inverse problem of recovering part of the domain boundary from boundary measurements of solutions to the Laplace equation on an accessible part of the boundary.     

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.     

High order convergence for collocation of second kind boundary integral equations on polygons

We propose collocation methods with smoothest splines to solve the integral equation of the second kind on a plane polygon. They are based on the bijectivity of the double layer potential between spaces of Sobolev type with arbitrary high regularity and involving the singular functions generated by the corners. If splines of order are used, we get quasi-optimal estimates in -norm and optimal order convergence for the -norm if . Numerical experiments are presented. Received November 20, 1996 / Accepted March 10, 1997     

Adaptive boundary element methods for strongly elliptic integral equations

Summary Integral operators are nonlocal operators. The operators defined in boundary integral equations to elliptic boundary value problems, however, are pseudo-differential operators on the boundary and, therefore, provide additional pseudolocal properties. These allow the successful application of adaptive procedures to some boundary element methods. In this paper we analyze these methods for general strongly elliptic integral equations and obtain a-posteriori error estimates for boundary element solutions. We also apply these methods to nodal collocation with odd degree splines. Some numerical examples show that these adaptive procedures are reliable and effective.This work was carried out while Dr. De-hao Yu was an Alexander-von-Humboldt-Stiftung research fellow at the University of Stuttgart in 1987, 1988     

Transmission problems and spectral theory for singular integral operators on Lipschitz domains

We prove the well-posedness of the transmission problem for the Laplacian across a Lipschitz interface, with optimal non-tangential maximal function estimates, for data in Lebesgue and Hardy spaces on the boundary. As a corollary, we show that the spectral radius of the (adjoint) harmonic double layer potential K∗ in is less than , whenever is a bounded convex domain and 1<p?2.     

Preconditioning techniques for the iterative solution of scattering problems

We consider a time-harmonic electromagnetic scattering problem for an inhomogeneous medium. Some symmetry hypotheses on the refractive index of the medium and on the electromagnetic fields allow to reduce this problem to a two-dimensional scattering problem. This boundary value problem is defined on an unbounded domain, so its numerical solution cannot be obtained by a straightforward application of usual methods, such as for example finite difference methods, and finite element methods. A possible way to overcome this difficulty is given by an equivalent integral formulation of this problem, where the scattered field can be computed from the solution of a Fredholm integral equation of second kind. The numerical approximation of this problem usually produces large dense linear systems. We consider usual iterative methods for the solution of such linear systems, and we study some preconditioning techniques to improve the efficiency of these methods. We show some numerical results obtained with two well known Krylov subspace methods, i.e., Bi-CGSTAB and GMRES.     

Nyström type methods for Fredholm integral equations with weak singularities

Nyström type methods are constructed and justified for a class of Fredholm integral equations of the second kind with kernels which may have weak diagonal and boundary singularities. The proposed approach is based on a suitable smoothing change of variables and product integration techniques. Global convergence estimates are derived and a collection of numerical results is given.     

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.     

The Galerkin scheme for Lavrentiev’sm-times iterated method to solve linear accretive Volterra integral equations of the first kind

In this paper, for the numerical solution of linear accretive Volterra integral equations of the first kind in Hilbert spaces we consider the Galerkin scheme for Lavrentiev’sm-times iterated method, i.e., for each parameter choice for Lavrentiev’sm-times iterated method the arisingm stabilized equations are discretized by the Galerkin scheme. An associated discrepancy principle as parameter choice strategy for this finite-dimensional version of Lavrentiev’sm-times iterated method is proposed, and corresponding convergence results are provided.     

The Laguerre spectral method for solving Neumann boundary value problems

In this paper, we propose a Laguerre spectral method for solving Neumann boundary value problems. This approach differs from the classical spectral method in that the homogeneous boundary condition is satisfied exactly. Moreover, a tridiagonal matrix is employed, instead of the full stiffness matrix encountered in the classical variational formulation of such problems. For analyzing the numerical errors, some basic results on Laguerre approximations are established. The convergence is proved. The numerical results demonstrate the efficiency of this approach.