Iterated Collocation Method Research Articles

Use of extrapolation for improving the order of convergence of eigenelement approximations

We consider the approximation of the eigenelements of a compact integral operator defined on C[0, 1] with a smooth kernel. We use the iterated collocation method based on r Gauss points and piecewise polynomials of degree ≤r − 1 on each subinterval of a nonuniform partition of [0, 1]. We obtain asymptotic expansions for the arithmetic means of m eigenvalues and also for the associated spectral projections. Using Richardson extrapolation, we show that the order of convergence O(h2r) in the iterated collocation method can be improved to O(h2r+2). Similar results hold for the Nyström method and for the iterated Galerkin method. We illustrate the improvement in the order of convergence by numerical experiments.

Iterated Collocation Methods for Volterra Integral Equations with Delay Arguments

In this paper we give a complete analysis of the global convergence and local superconvergence properties of piecewise polynomial collocation for Volterra integral equations with constant delay. This analysis includes continuous collocation-based Volterra-Runge-Kutta methods as well as iterated collocation methods and their discretizations.

Iterated collocation methods for Volterra integral equations with delay arguments

In this paper we give a complete analysis of the global convergence and local superconvergence properties of piecewise polynomial collocation for Volterra integral equations with constant delay. This analysis includes continuous collocation-based Volterra-Runge-Kutta methods as well as iterated collocation methods and their discretizations.

Asymptotic error expansion of a collocation-type method for Volterra-Hammerstein integral equations

Recently, Kumar and Sloan introduced a new collocation-type method for the numerical solution of Fredholm-Hammerstein integral equations. Brunner applied this method to nonlinear Volterra integral and integro-differential equations and discussed its connection with the iterated collocation method. In the present paper, the asymptotic error expansion of this method for nonlinear Volterra integral equations at mesh points is obtained. We show that when piecewise polynomials of Θin p-1 are used, the approximate solution admits an error expansion in powers of stepsize h, beginning with a term h p . For a special choice of the collocation points, the leading terms in the error expansion for the collocation solution contain only even powers of the stepsize h, beginning with a term h 2 p . Thus Richardson's extrapolation can be performed on the solution; and this will greatly increase the accuracy of the numerical solution.

Numerical analysis of boundary integral solution of the Helmholtz equation in domains with non-smooth boundaries

In the paper we carry out a complete analysis of several efficient numerical methods for the solution of boundary integral equations defined on a non-smooth boundary. In particular the solution of the Helmholtz equation in the exterior of a closed wedge is studied. The analytical behaviour of the solution of the resulting boundary integral equation (with a non-compact operator) near the wedge is investigated. Numerical analysis of the collocation and iterated collocation method for the problem is presented. Graded meshes are used to reflect the «singular» behaviour of the analytical solution, as well as the degree of the polynomial approximant, in order to yield results with «optimal convergence rates»

Implicitly linear collocation methods for nonlinear Volterra equations

Recently, Kumar and Sloan introduced a new collocation-type method for numerical solution of Hammerstein integral equations. In the present paper we apply this method (which will be referred to as the implicitly linear collocation method) to nonlinear Volterra integral and integro-differential equations and discuss its connection with the iterated collocation method.

On Implicitly Linear and Iterated Collocation Methods for Hammerstein Integral Equations

On Implicitly Linear and Iterated Collocation Methods for Hammerstein Integral Equations

Extrapolation of the Iterated–Collocation Method for Integral Equations of the Second Kind

For integral equations of the second kind with appropriate smoothness of both kernel and exact solution, it is known that the iterated–collocation method based on discontinuous piecewise polynomials of order r, and on collocation at the Gauss points, can exhibit a uniform (superconvergent) error of order $2r$. This paper shows that the order of the iterated–collocation approximation on an arbitrary mesh can be further improved by one step of Richardson extrapolation, assuming the calculation to be repeated with each subinterval halved.

The Numerical Solution of Nonlinear Volterra Integral Equations of the Second Kind by Collocation and Iterated Collocation Methods

The subject of this paper is a variable stepsize one-step method of collocation type for solving general nonlinear second kind Volterra integral equations. We extend the iterated collocation method corresponding to polynomial spline collocation to nonlinear Volterra integral equations of the second kind. The resulting superconvergence properties of either the collocation approximation or the iterated collocation approximation are used to obtain (local and global) error estimates which in turn form the basis of a variable stepsize code. The performance of this code is illustrated by means of numerous test problems.

Discrete Collocation Methods for Second Kind Fredholm Integral Equations

The numerical solution of second kind Fredholm integral equations by the discrete collocation method and a discrete iterated collocation method is studied. Orders of convergence for the two discrete methods are given for the case in which an interpolatory quadrature rule of appropriate precision is used. If the kernel is sufficiently smooth, these orders of convergence will be the same as those of the exact methods. Some numerical results are given.

Iterated Collocation Methods and Their Discretizations for Volterra Integral Equations

It is known that the numerical solution of Volterra integral equations of the second kind by polynomial spline collocation at the Gauss–Legendre points does not lead to local superconvergence at the knots of the approximating function. In the present paper we show that iterated collocation approximation restores optimal local superconvergence at the knots but does not yield global superconvergence on the entire interval of integration, in contrast to Fredholm integral equations with smooth kernels. We also analyze the discretized versions (obtained by suitable numerical quadrature) of the collocation and iterated collocation methods.

Collocation methods for two dimensional weakly singular integral equations

AbstractGiven a Fredhoim integral equation of the second kind, which is defined over a certain region ⊆ R2, we define and , two different numerical approximations to its solution, using the collocation and iterated collocation methods respectively. We describe without proof some known results concerning the general convergence properties of and when the kernel and solution of the integral equation are smooth. Then, we prove rigorously order of convergence estimates for and which are applicable in the practically siginificant case when is a rectangle, and the kernel of the integral equation is weakly singular. These estimates are illustrated by the numerical solution of a two dimensional weakly singular equation which arises in electrical engineering.