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.     

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.     

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.     

Analysis of general quadrature methods for integral equations of the second kind

Summary This paper is concerned with a class of approximation methods for integral equations of the form , wherea andb are finite,f andy are continuous and the kernelk may be weakly singular. The methods are characterized by approximate equations of the form ; such methods include the Nyström method and a variety of product-integration methods. A general convergence theory is developed for methods of this type. In suitable cases it has the feature that its application to a specific method depends only on a knowledge of convergence properties of the underlying quadrature rule. The theory is used to deduce convergence results, some of them new, for a number of specific methods.Work supported by the U.S. Department of Energy     

On multilevel iterative methods for integral equations of the second kind and related problems

Summary We describe a unifying framework for multigrid methods and projection-iterative methods for integral equations of the second kind, and for the iterative aggregation method for solving input-output relations. The methods are formulated as iterations combined with a defect correction in a subspace. Convergence proofs use contraction arguments and thus involve the nonlinear case automatically. Some new results are presented.     

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     

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.     

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     

Backward differentiation type formulas for Volterra integral equations of the second kind

Summary Numerical integration formulas are discussed which are obtained by differentiation of the Volterra integral equation and by applying backward differentiation formulas to the resulting integro-differential equation. In particular, the stability of the method is investigated for a class of convolution kernels. The accuracy and stability behaviour of the method proposed in this paper is compared with that of (i) a block-implicit Runge-Kutta scheme, and (ii) the scheme obtained by applying directly a quadrature rule which is reducible to the backward differentiation formulas. The present method is particularly advantageous in the case of stiff Volterra integral equations.     

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.     

Superconvergence of collocation methods for Volterra and Abel integral equations of the second kind

Summary This paper deals with the question of the attainable order of convergence in the numerical solution of Volterra and Abel integral equations by collocation methods in certain piecewise polynomial spaces and which are based on suitable interpolatory quadrature for the resulting moment integrals. The use of a (nonlinear) variation of constants formula for the representation of the error function in terms of the defect allows for a unified treatment of equations with continuous and weakly singular kernels.     

Superconvergence of piecewise polynomial Galerkin approximations,for Fredholm integral equations of the second kind

Summary Piecewise polynomial Galerkin approximations for Fredholm integral equations of the second kind are shown to posses superconvergence properties in some circumstances.     

Asymptotic expansions for multistep methods applied to nonlinear Volterra integral equations of the second kind

Summary This paper deals with linear multistep methods applied to nonlinear, nonsingular Volterra integral equations of the second kind. Analogously to the theory of W.B. Gragg, the existence of asymptotic expansions in the stepsizeh is proved. Under certain conditions only even powers ofh occur. As a special case, the midpoint rule is treated, a short numerical example for the applicability to extrapolation techniques is given.     

Multigrid methods for boundary integral equations

Summary Multigrid methods are applied for solving algebraic systems of equations that occur to the numerical treatment of boundary integral equations of the first and second kind. These methods, originally formulated for partial differential equations of elliptic type, combine relaxation schemes and coarse grid corrections. The choice of the relaxation scheme is found to be essential to attain a fast convergent iterative process. Theoretical investigations show that the presented relaxation scheme provides a multigrid algorithm of which the rate of convergence increases with the dimension of the finest grid. This is illustrated for the calculation of potential flow around an aerofoil.     

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.     

Natural Continuous Extensions of Runge-Kutta methods for Volterra integrodifferential equations

Summary In this paper we deal with a very general class of Runge-Kutta methods for the numerical solution of Volterra integrodifferential equations. Our main contribution is the development of the theory of Natural Continuous Extensions (NCEs), i.e. piecewice polynomial functions which interpolate the values given by the RK-method at the mesh points. The particular features of these NCEs allow us to construct tail approximations which are quite efficient since they require a minimal number of kernel evaluations.     

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 of a spline collocation method for strongly elliptic multidimensional singular integral equations

Summary We consider a spline collocation method for strongly elliptic zero order pseudodifferential equationsp _gw Au=f on a cube =(0, 1) ^m . Utilizing multilinear spline functions which are zero at the boundary we collocate at the meshpoints inside . For classical strongly elliptic translation invariant pseudodifferential operators, we verify the stability of the considered collocation method inL ²(). Afterwards, form2 and a right hand sidefH ⁸(),s>m/2, we prove an asymptotic convergence estimate.The author has been supported by a grant of Deutsche Forschungsgemeinschaft under grant number Ko 634/32-1     

On the fast matrix multiplication in the boundary element method by panel clustering

Summary The boundary element method (BEM) leads to a system of linear equations with a full matrix, while FEM yields sparse matrices. This fact seems to require much computational work for the definition of the matrix, for the solution of the system, and, in particular, for the matrix-vector multiplication, which always occurs as an elementary. In this paper a method for the approximate matrix-vector multiplication is described which requires much less arithmetical work. In addition, the storage requirements are strongly reduced.This paper reports results of a research project supported by the DFG (Schwerpunktprogramm Finite Approximationen in der Strömungsmechanik     

Discretization of Volterra integral equations of the first kind (II)

Summary In the present paper integral equations of the first kind associated with strictly monotone Volterra integral operators are solved by projecting the exact solution of such an equation into the spaceS _m ^(–1) (Z _N ) of piecewise polynomials of degreem0, possessing jump discontinuities on the setZ _N of knots. Since the majority of direct one-step methods (including the higher-order block methods) result from particular discretizations of the moment integrals occuring in the above projection method we obtain a unified convergence analysis for these methods; in addition, the above approach yields the tools to deal with the question of the connection between the location of the collocation points used to determine the projection inS _m ^(–1) (Z _N ) and the order of convergence of the method.This research was supported by the National Research Council of Canada (Grant No. A-4805)