Spectral collocation method for compact integral operators

We propose and analyze a spectral collocation method for integral equations with compact kernels, e.g. piecewise smooth kernels and weakly singular kernels of the form $\frac{1}{|t-s|^\mu}, \; 0\u3c\mu\u3c1.$ We prove that 1) for integral equations, the convergence rate depends on the smoothness of true solutions $y(t)$. If $y(t)$ satisfies condition (R): $\|y^{(k)}\|_{L^\infty[0,T]}\leq ck!R^{-k}$}, we obtain a geometric rate of convergence; if $y(t)$ satisfies condition (M): $\|y^{(k)}\|_{L^{\infty}[0,T]}\leq cM^k$, we obtain supergeometric rate of convergence for both Volterra equations and Fredholm equations and related integro differential equations; 2) for eigenvalue problems, the convergence rate depends on the smoothness of eigenfunctions. The same convergence rate for the largest modulus eigenvalue approximation can be obtained. Moreover, the convergence rate doubles for positive compact operators. Our numerical experiments confirm our theoretical results

A Product Integration type Method for solving Nonlinear Integral Equations in L

This paper deals with nonlinear Fredholm integral equations of the second kind. We study the case of a weakly singular kernel and we set the problem in the space L 1 ([a, b], C). As numerical method, we extend the product integration scheme from C 0 ([a, b], C) to L 1 ([a, b], C)

ANALYSIS OF ITERATIVE METHODS FOR THE SOLUTION OF BOUNDARY INTEGRAL EQUATIONS WITH APPLICATIONS TO THE HELMHOLTZ PROBLEM

This thesis is concerned with the numerical solution of boundary integral equations and the numerical analysis of iterative methods. In the first part, we assume the boundary to be smooth in order to work with compact operators; while in the second part we investigate the problem arising from allowing piecewise smooth boundaries. Although in principle most results of the thesis apply to general problems of reformulating boundary value problems as boundary integral equations and their subsequent numerical solutions, we consider the Helmholtz equation arising from acoustic problems as the main model problem. In Chapter 1, we present the background material of reformulation of Helmhoitz boundary value problems into boundary integral equations by either the indirect potential method or the direct method using integral formulae. The problem of ensuring unique solutions of integral equations for exterior problems is specifically discussed. In Chapter 2, we discuss the useful numerical techniques for solving second kind integral equations. In particular, we highlight the superconvergence properties of iterated projection methods and the important procedure of Nystrom interpolation. In Chapter 3, the multigrid type methods as applied to smooth boundary integral equations are studied. Using the residual correction principle, we are able to propose some robust iterative variants modifying the existing methods to seek efficient solutions. In Chapter 4, we concentrate on the conjugate gradient method and establish its fast convergence as applied to the linear systems arising from general boundary element equations. For boundary integral equalisations on smooth boundaries we have observed, as the underlying mesh sizes decrease, faster convergence of multigrid type methods and fixed step convergence of the conjugate gradient method. In the case of non-smooth integral boundaries, we first derive the singular forms of the solution of boundary integral solutions for Dirichlet problems and then discuss the numerical solution in Chapter 5. Iterative methods such as two grid methods and the conjugate gradient method are successfully implemented in Chapter 6 to solve the non-smooth integral equations. The study of two grid methods in a general setting and also much of the results on the conjugate gradient method are new. Chapters 3, 4 and 5 are partially based on publications [4], [5] and [35] respectively.Department of Mathematics and Statistics, Polytechnic South Wes

Quadrature methods for 2D and 3D problems

AbstractIn this paper we give an overview on well-known stability and convergence results for simple quadrature methods based on low-order composite quadrature rules and applied to the numerical solution of integral equations over smooth manifolds. First, we explain the methods for the case of second-kind equations. Then we discuss what is known for the analysis of pseudodifferential equations. We explain why these simple methods are not recommended for integral equations over domains with dimension higher than one. Finally, for the solution of a two-dimensional singular integral equation, we prove a new result on a quadrature method based on product rules

Boundary integral methods in high frequency scattering

In this article we review recent progress on the design, analysis and implementation of numerical-asymptotic boundary integral methods for the computation of frequency-domain acoustic scattering in a homogeneous unbounded medium by a bounded obstacle. The main aim of the methods is to allow computation of scattering at arbitrarily high frequency with finite computational resources

The numerical solution of the dynamic fluid-structure interaction problem.

Merged with duplicate record 10026.1/2055 on 12.04.2017 by CS (TIS)In this thesis we consider the problem of the dynamic fluid-structure interaction between a finite elastic structure and the acoustic field in an unbounded fluid-filled exterior domain. We formulate the exterior acoustic problem as an integral equation over the structure surface. However, the classical boundary integral equation formulations of this problem do not have unique solutions at certain characteristic frequencies (which depend on the surface) and it is necessary to employ modified boundary integral equation formulations which are valid for all frequencies. The modified integral equation formulation used here involves certain arbitrary parameters and we shall study the effect of these parameters on the stability and accuracy of the numerical methods used to solve the integral equation. We then couple the boundary element analysis of the exterior acoustic problem with a finite element analysis of the elastic structure to investigate the interaction between the structure and the acoustic field. Recently there has been some controversy over whether or not the coupled problem suffers from the non-uniqueness problems associated with the classical integral equation formulations of the exterior acoustic problem. We resolve this question by demonstrating that the solution to the coupled problem is not unique at the characteristic frequencies and that we need to employ an integral equation formulation valid for all frequencies. We discuss the accuracy of our numerical results for both the acoustic problem and the coupled problem, for a number of axisymmetric and fully three-dimensional problems. Finally, we apply our method to the problem of a piezoelectric sonar transducer transmitting an acoustic signal in water, and observe reasonable agreement between our theoretical predictions and some experimental results.Admiralty Research Establishment, Portlan