The convergence of spline collocation for strongly elliptic equations on curves

Summary Most boundary element methods for two-dimensional boundary value problems are based on point collocation on the boundary and the use of splines as trial functions. Here we present a unified asymptotic error analysis for even as well as for odd degree splines subordinate to uniform or smoothly graded meshes and prove asymptotic convergence of optimal order. The equations are collocated at the breakpoints for odd degree and the internodal midpoints for even degree splines. The crucial assumption for the generalized boundary integral and integro-differential operators is strong ellipticity. Our analysis is based on simple Fourier expansions. In particular, we extend results by J. Saranen and W.L. Wendland from constant to variable coefficient equations. Our results include the first convergence proof of midpoint collocation with piecewise constant functions, i.e., the panel method for solving systems of Cauchy singular integral equations.Dedicated to Prof. Dr. Dr. h.c. mult. Lothar Collatz on the occasion of his 75th birthdayThis work was begun at the Technische Hochschule Darmstadt where Professor Arnold was supported by a North Atlantic Treaty Organization Postdoctoral Fellowship. The work of Professor Arnold is supported by NSF grant BMS-8313247. The work of Professor Wendland was supported by the Stiftung Volkswagenwerk     

Continuous time collocation methods for Volterra-Fredholm integral equations

Summary Integral equations of mixed Volterra-Fredholm type arise in various physical and biological problems. In the present paper we study continuous time collocation, time discretization and their global and discrete convergence properties.     

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     

The spline collocation method for parabolic boundary integral equations on smooth curves

Summary. We consider the spline collocation method for a class of parabolic pseudodifferential operators. We show optimal order convergence results in a large scale of anisotropic Sobolev spaces. The results cover the classical boundary integral equations for the heat equation in the general case where the spatial domain has a smooth boundary in the plane. Our proof is based on a localization technique for which we use our recent results proved for parabolic pseudodifferential operators. For the localization we need also some special spline approximation results in anisotropic Sobolev spaces. Received May 17, 2001 / Revised version received February 19, 2002 / Published online April 17, 2002     

On Bateman's method for second kind integral equations

Summary Under suitable conditions, we prove the convergence of the Bateman method for integral equations defined over bounded domains inR ^d ,d1. The proof makes use of Hilbert space methods, and requires the integral operator to be non-negative definite. For one-dimensional integral equations over finite intervals, estimated rates of convergence are obtained which depend on the smoothness of the kernel, but are independent of the inhomogeneous term. In particular, for aC kernel andn reasonably spaced Bateman points, the convergence is shown to be faster than any power of 1/n. Numerical calculations support this result.     

Spline collocation for singular integro-differential equations over (0.1)

Summary This paper analyses the convergence of spline collocation methods for singular integro-differential equations over the interval (0.1). As trial functions we utilize smooth polynomial splines the degree of which coincides with the order of the equation. Depending on the choice of collocation points we obtain sufficient and even necessary conditions for the convergence in sobolev norms. We give asymptotic error estimates and some numerical results.     

Stability and robustness of collocation methods for Abel-type integral equations

Summary We study stability aspects of collocation methods for Abel-type integral equations of the first kind using piecewise polynomials. These collocation methods may be formulated as projection methods. Stability is defined as the boundedness of the sequence of projectors in their natural setting. Robustness is essentially the optimal asymptotic insensitivity to perturbations in the data. We show that stability and robustness are equivalent for the above collocation methods. This allows us to obtain optimal error estimates for some methods that are well-known to be robust. We also present numerical results for some methods which appear to be robust.Research supported in part by the United States Army under Contract No. DAAG 29-83-K-0109     

On spline Galerkin methods for singular integral equations with piecewise continuous coefficients

Summary This article analizes the convergence of the Galerkin method with polynomial splines on arbitrary meshes for systems of singular integral equations with piecewise continuous coefficients inL ² on closed or open Ljapunov curves. It is proved that this method converges if and, for scalar equations and equidistant partitions, only if the integral operator is strongly elliptic (in some generalized sense). Using the complete asymptotics of the solution, we provide error estimates for equidistant and for special nonuni-form partitions.     

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 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.     

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.     

Duality estimates for the numerical solution of integral equations

Summary We formulate and prove Aubin-Nitsche-type duality estimates for the error of general projection methods. Examples of applications include collocation methods and augmented Galerkin methods for boundary integral equations on plane domains with corners and three-dimensional screen and crack problems. For some of these methods, we obtain higher order error estimates in negative norms in cases where previous formulations of the duality arguments were not applicable.     

Fast solution methods for fredholm integral equations of the second kind

Summary The main purpose of this paper is to describe a fast solution method for one-dimensional Fredholm integral equations of the second kind with a smooth kernel and a non-smooth right-hand side function. Let the integral equation be defined on the interval [–1, 1]. We discretize by a Nyström method with nodes {cos(j/N)} _j ^=0/N . This yields a linear system of algebraic equations with an (N+1)×(N+1) matrixA. GenerallyN has to be chosen fairly large in order to obtain an accurate approximate solution of the integral equation. We show by Fourier analysis thatA can be approximated well by , a low-rank modification of the identity matrix. ReplacingA by in the linear system of algebraic equations yields a new linear system of equations, whose elements, and whose solution , can be computed inO (N logN) arithmetic operations. If the kernel has two more derivatives than the right-hand side function, then is shown to converge optimally to the solution of the integral equation asN increases.We also consider iterative solution of the linear system of algebraic equations. The iterative schemes use bothA andÃ. They yield the solution inO (N ²) arithmetic operations under mild restrictions on the kernel and the right-hand side function.Finally, we discuss discretization by the Chebyshev-Galerkin method. The techniques developed for the Nyström method carry over to this discretization method, and we develop solution schemes that are faster than those previously presented in the literature. The schemes presented carry over in a straightforward manner to Fredholm integral equations of the second kind defined on a hypercube.     

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     

A quadrature-based approach to improving the collocation method

Summary The collocation method is a popular method for the approximate solution of boundary integral equations, but typically does not achieve the high order of convergence reached by the Galerkin method in appropriate negative norms. In this paper a quadrature-based method for improving upon the collocation method is proposed, and developed in detail for a particular case. That case involves operators with even symbol (such as the logarithmic potential) operating on 1-periodic functions; a smooth-spline trial space of odd degree, with constant mesh spacingh=1/n; and a quadrature rule with 2n points (where ann-point quadrature rule would be equivalent to the collocation method). In this setting it is shown that a special quadrature rule (which depends on the degree of the splines and the order of the operator) can yield a maximum order of convergence two powers ofh higher than the collocation method.     

Convergence analysis of discretization methods for nonlinear first kind Volterra integral equations

Summary An existence and uniqueness result is given for nonlinear Volterra integral equations of the first kind. This permits, by means of analogous discrete manipulations, a general convergence analysis for a wide class of discretization methods for nonlinear first kind Volterra integral equations to be presented. A concept of optimal consistency allows twosided error bounds to be derived.     

On monotone Abel-Volterra integral equations on the half line

Summary In this paper we reanalyze the trapezoidal method for the solution of nonlinear Abel-Volterra integral equations on the half line. We prove the convergence of the method in the uniform norm, provided the nonlinearity is Lipschitz-continuous and strictly monotone.Research supported in part by the United States Army under contracts DAAG29-83-K-0109 and DAAG 29-85-G-0009     

L p estimates of boundary integral equations for some nonlinear boundary value problems

Summary Recently, Galerkin and collocation methods have been analyzed for boundary integral equation formulations of some potential problems in the plane with nonlinear boundary conditions, and stability results and error estimates in theH ^1/2-norm have been proved (Ruotsalainen and Wendland, and Ruotsalainen and Saranen). We show that these results extend toL ^p setting without any extra conditions. These extensions are proved by studying the uniform boundedness of the inverses of the linearized integral operators, and then considering the nonlinear equations. The fact that inH ^1/2 setting the nonlinear operator is a homeomorphism with Lipschitz continuous inverse plays a crucial role. Optimal error estimates for the Galerkin and collocation method inL ^p space then follow.This research was performed while the second author was visiting professor at the University of Delaware, spring 1989     

Numerical solution of integral equations,fast algorithms and Krein-Sobolev equation

Summary This paper contains a unified rigorous approach for the treatment of fast numerical algorithms for different classes of Fredholm integral equations of second kind. The Krein-Sobolev functional-differential nonlinear equation for the resolvent provides the ground for a unified approach. New results concerning the general analysis of the Krein-Sobolev equation and the convergence and stability of related numerical schemes are also presented.     

Numerical solution of cauchy-type integral equations of index −1 by collocation methods

This paper describes a collocation method for solving constant coefficient Cauchy-type singular integral equations of index –1. A technique for reducing the set of linear equations resulting from collocation to match the number of unknows is described. The uniform convergence analysis of the resulting method is presented and convergence rates based on the smoothness of the data are given.This work was partially supported by CNPq under grant #300105/88-6(RN).