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.     

The Superconvergence of the Composite Trapezoidal Rule for Hadamard Finite Part Integrals

The composite trapezoidal rule has been well studied and widely applied for numerical integrations and numerical solution of integral equations with smooth or weakly singular kernels. However, this quadrature rule has been less employed for Hadamard finite part integrals due to the fact that its global convergence rate for Hadamard finite part integrals with (p+1)-order singularity is p-order lower than that for the Riemann integrals in general. In this paper, we study the superconvergence of the composite trapezoidal rule for Hadamard finite part integrals with the second-order and the third-order singularity, respectively. We obtain superconvergence estimates at some special points and prove the uniqueness of the superconvergence points. Numerical experiments confirm our theoretical analysis and show that the composite trapezoidal rule is efficient for Hadamard finite part integrals by noting the superconvergence phenomenon. The work of this author was partially supported by the National Natural Science Foundation of China(No.10271019), a grant from the Research Grants Council of the Hong Kong Special Administractive Region, China (Project No. City 102204) and a grant from the Laboratory of Computational Physics The work of this author was supported in part by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 102204).     

Explicit expressions for three-dimensional boundary integrals in linear elasticity

On employing isoparametric, piecewise linear shape functions over a flat triangle, exact formulae are derived for all surface potentials involved in the numerical treatment of three-dimensional singular and hyper-singular boundary integral equations in linear elasticity. These formulae are valid for an arbitrary source point in space and are represented as analytical expressions along the edges of the integration triangle. They can be employed to solve integral equations defined on triangulated surfaces via a collocation method or may be utilized as analytical expressions for the inner integrals in a Galerkin technique. A numerical example involving a unit triangle and a source point located at various distances above it, as well as sample problems solved by a collocation boundary element method for the Lamé equation are included to validate the proposed formulae.     

Numerical integration of functions with a very small significant support

In some applications, one has to deal with the problem of integrating, over a bounded interval, a smooth function taking significant values, with respect to the machine precision or to the accuracy one wants to achieve, only in a very small part of the domain of integration. In this paper, we propose a simple and efficient numerical approach to compute or discretize integrals of this type. We also consider a class of second kind integral equations whose integral operator has the above behavior. Some numerical testing is presented.     

Integration of singular Galerkin-type boundary element integrals for 3D elasticity problems

Summary. A Galerkin approximation of both strongly and hypersingular boundary integral equation (BIE) is considered for the solution of a mixed boundary value problem in 3D elasticity leading to a symmetric system of linear equations. The evaluation of Cauchy principal values (v. p.) and finite parts (p. f.) of double integrals is one of the most difficult parts within the implementation of such boundary element methods (BEMs). A new integration method, which is strictly derived for the cases of coincident elements as well as edge-adjacent and vertex-adjacent elements, leads to explicitly given regular integrand functions which can be integrated by the standard Gauss-Legendre and Gauss-Jacobi quadrature rules. Problems of a wide range of integral kernels on curved surfaces can be treated by this integration method. We give estimates of the quadrature errors of the singular four-dimensional integrals. Received June 25, 1995 / Revised version received January 29, 1996     

The numerical evaluation of Hadamard finite-part integrals

Summary A quadrature rule is described for the numerical evaluation of Hadamard finite-part integrals with a double pole singularity within the range of integration. The rule is based upon the observation that such an integral is the derivative of a Cauchy principal value integral.     

Truncated interpolation processes on unbounded intervals and their applications

The authors describe the major results they have recently obtained on the truncation of interpolation processes on unbounded intervals, and on their applications to the numerical evaluation of corresponding integrals and to the resolution of a class of integral equations. Work partially supported by Progetto Innovativo GNCS “Trattamento numerico di equazioni integrali e connessi problemi di approssimazione e quadratura”.     

Hybrid Galerkin boundary elements: theory and implementation

Summary. In this paper we present a new quadrature method for computing Galerkin stiffness matrices arising from the discretisation of 3D boundary integral equations using continuous piecewise linear boundary elements. This rule takes as points some subset of the nodes of the mesh and can be used for computing non-singular Galerkin integrals corresponding to pairs of basis functions with non-intersecting supports. When this new rule is combined with standard methods for the singular Galerkin integrals we obtain a “hybrid” Galerkin method which has the same stability and asymptotic convergence properties as the true Galerkin method but a complexity more akin to that of a collocation or Nystr?m method. The method can be applied to a wide range of singular and weakly-singular first- and second-kind equations, including many for which the classical Nystr?m method is not even defined. The results apply to equations on piecewise-smooth Lipschitz boundaries, and to non-quasiuniform (but shape-regular) meshes. A by-product of the analysis is a stability theory for quadrature rules of precision 1 and 2 based on arbitrary points in the plane. Numerical experiments demonstrate that the new method realises the performance expected from the theory. Received January 22, 1998 / Revised version received May 26, 1999 / Published online April 20, 2000 –? Springer-Verlag 2000     

Sigmoidal transformations and the Euler-Maclaurin expansion for evaluating certain Hadamard finite-part integrals

Summary. Starting with some results of Lyness concerning the Euler-Maclaurin expansion of Cauchy principal value integrals over it is shown how, by the use of sigmoidal transformations, good approximations can be found for the Hadamard finite-part integral where The analysis is illustrated by some numerical examples. Received March 13, 1996     

Semi-analytic integration of hypersingular Galerkin BIEs for three-dimensional potential problems

An accurate and efficient semi-analytic integration technique is developed for three-dimensional hypersingular boundary integral equations of potential theory. Investigated in the context of a Galerkin approach, surface integrals are defined as limits to the boundary and linear surface elements are employed to approximate the geometry and field variables on the boundary. In the inner integration procedure, all singular and non-singular integrals over a triangular boundary element are expressed exactly as analytic formulae over the edges of the integration triangle. In the outer integration scheme, closed-form expressions are obtained for the coincident case, wherein the divergent terms are identified explicitly and are shown to cancel with corresponding terms from the edge-adjacent case. The remaining surface integrals, containing only weak singularities, are carried out successfully by use of standard numerical cubatures. Sample problems are included to illustrate the performance and validity of the proposed algorithm.     

Computing multiple integrals involving matrix exponentials

In this paper, a generalization of a formula proposed by Van Loan [Computing integrals involving the matrix exponential, IEEE Trans. Automat. Control 23 (1978) 395–404] for the computation of multiple integrals of exponential matrices is introduced. In this way, the numerical evaluation of such integrals is reduced to the use of a conventional algorithm to compute matrix exponentials. The formula is applied for evaluating some kinds of integrals that frequently emerge in a number classical mathematical subjects in the framework of differential equations, numerical methods and control engineering applications.     

Systems of Volterra integral equations with many-dimensional partial integrals

The unique solvability of systems of linear and nonlinear Volterra integral equations with partial integrals is studied in this paper. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 53, Suzdal Conference-2006, Part 1, 2008.     

Discrete boundary element methods on general meshes in 3D

Abstract. This paper is concerned with the stability and convergence of fully discrete Galerkin methods for boundary integral equations on bounded piecewise smooth surfaces in . Our theory covers equations with very general operators, provided the associated weak form is bounded and elliptic on , for some . In contrast to other studies on this topic, we do not assume our meshes to be quasiuniform, and therefore the analysis admits locally refined meshes. To achieve such generality, standard inverse estimates for the quasiuniform case are replaced by appropriate generalised estimates which hold even in the locally refined case. Since the approximation of singular integrals on or near the diagonal of the Galerkin matrix has been well-analysed previously, this paper deals only with errors in the integration of the nearly singular and smooth Galerkin integrals which comprise the dominant part of the matrix. Our results show how accurate the quadrature rules must be in order that the resulting discrete Galerkin method enjoys the same stability properties and convergence rates as the true Galerkin method. Although this study considers only continuous piecewise linear basis functions on triangles, our approach is not restricted in principle to this case. As an example, the theory is applied here to conventional “triangle-based” quadrature rules which are commonly used in practice. A subsequent paper [14] introduces a new and much more efficient “node-based” approach and analyses it using the results of the present paper. Received December 10, 1997 / Revised version received May 26, 1999 / Published online April 20, 2000 –? Springer-Verlag 2000     

Local error estimates in quadrature

A theoretical error estimate for quadrature formulas, which depends on four approximations of the integral, is derived. We obtain a bound, often sharper than the trivial one, which requires milder conditions to be satisfied than a similar result previously presented by Laurie. A selection of numerical tests with one-dimensional integrals is reported, to show how the error estimate works in practice.     

Solving a class of two-dimensional linear and nonlinear Volterra integral equations by the differential transform method

In this paper, we will give some results for developing the two-dimensional differential transform (TDDT) for double integrals. Then the TDDT method will be developed for solving a class of two-dimensional linear and nonlinear Volterra integral equations. We also give some examples to demonstrate the accuracy of the method.     

Reducible quadrature methods for Volterra integral equations of the first kind

Quadrature rules, generated by linear multistep methods for ordinary differential equations, are employed to construct a wide class of direct quadrature methods for the numerical solution of first kind Volterra integral equations. Our class covers several methods previously considered in the literature. The methods are convergent provided that both the first and second characteristic polynomial of the linear multistep method satisfy the root condition. Furthermore, the stability behaviour for fixed positive values of the stepsizeh is analyzed, and it turns out that convergence implies (fixedh) stability. The subclass formed by the backward differentiation methods up to order six is discussed and illustrated with numerical examples.     

Methods for the rapid solution of the pricing PIDEs in exponential and Merton models

The pricing equations for options on assets that follow jump-diffusion processes contain integrals in addition to the usual differential terms. These integrals usually make such equations expensive to solve numerically. Although Fast Fourier Transform methods can be used to to evaluate the integrals at all mesh points simultaneously, they are costly since the computational region must be extended in order to avoid problems with wrap around. Other numerical difficulties arise when the density function for the jump size is not smooth, as in the Kou double exponential model. We present new solution methods which are based on the fact that even when the problems contain time-dependent parameters the integrals often satisfy easily solved ordinary or parabolic partial differential equations. In particular, we show that by using the operator splitting method proposed by Andersen and Andreasen it is possible to reduce the solution of the pricing equation in the Kou and similar models to a sequence of ordinary differential equations at each time step. We discuss the methods and present results of numerical experiments.     

On the uniform convergence of Gaussian quadrature rules for Cauchy principal value integrals

Summary In a previous paper the authors proposed a modified Gaussian rule _* ^m (wf;t)to compute the integral (wf;t) in the Cauchy principal value sense associated with the weightw, and they proved the convergence in closed sets contained in the integration interval. The main purpose of the present work is to prove uniform convergence of the sequence { _* ^m (wf;t)} on the whole integration interval and to give estimates for the remainder term. The same results are shown for particular subsequences of the Gaussian rules _m (wf;t) for the evaluation of Cauchy principal value integrals. A result on the uniform convergence of the product rules is also discussed and an application to the numerical solution of singular integral equations is made.     

The numerical evaluation of two integral transforms

We derive recurrence relationships for the evaluation of two integral transforms which are of interest for the numerical solution of some integral equations and for the construction of certain quadrature rules.     

Estimating quadrature errors for an efficient method for quasi-singular boundary integral

The numerical resolution of the boundary integral equations applied to the differential equations of Laplace, Helmholtz and Maxwell requires the handling of quasi-singular integrals with different order of singularity. The numerical approximation of the integral equations of different kinds is made by boundary finite elements. In this paper, we present a complete survey for estimating quadrature errors for the numerical techniques proposed by Huang and Cruse [Q. Huang, T.A. Cruse, Some notes on singular integral techniques in boundary element analysis, Int. J. Numer. Methods Eng. 36 (15) (1993) 2643-2659], to calculate the quasi-singular integrals. To validate the accuracy and efficiency of these techniques and approve our study some numerical examples are presented and discussed.