The superconvergence of the composite midpoint rule for the finite-part integral

The composite midpoint rule is probably the simplest one among the Newton-Cotes rules for Riemann integral. However, this rule is divergent in general for Hadamard finite-part integral. In this paper, we turn this rule to a useful one and, apply it to evaluate Hadamard finite-part integral as well as to solve the relevant integral equation. The key point is based on the investigation of its pointwise superconvergence phenomenon, i.e., when the singular point coincides with some a priori known point, the convergence rate of the midpoint rule is higher than what is globally possible. We show that the superconvergence rate of the composite midpoint rule occurs at the midpoint of each subinterval and obtain the corresponding superconvergence error estimate. By applying the midpoint rule to approximate the finite-part integral and by choosing the superconvergence points as the collocation points, we obtain a collocation scheme for solving the finite-part integral equation. More interesting is that the inverse of the coefficient matrix of the resulting linear system has an explicit expression, by which an optimal error estimate is established. Some numerical examples are provided to validate the theoretical analysis.     

A numerical scheme for a class of nonlinear Fredholm integral equations of the second kind

In this paper an iterative approach for obtaining approximate solutions for a class of nonlinear Fredholm integral equations of the second kind is proposed. The approach contains two steps: at the first one, we define a discretized form of the integral equation and prove that by considering some conditions on the kernel of the integral equation, solution of the discretized form converges to the exact solution of the problem. Following that, in the next step, solution of the discretized form is approximated by an iterative approach. We finally on some examples show the efficiency of the proposed approach.     

On superconvergence techniques

A brief survey with a bibliography of superconvergence phenomena in finding a numerical solution of differential and integral equations is presented. A particular emphasis is laid on superconvergent schemes for elliptic problems in the plane employing the finite element method.     

Superconvergence of the composite Simpson’s rule for a certain finite-part integral and its applications

The superconvergence phenomenon of the composite Simpson’s rule for the finite-part integral with a third-order singularity is studied. The superconvergence points are located and the superconvergence estimate is obtained. Some applications of the superconvergence result, including the evaluation of the finite-part integrals and the solution of a certain finite-part integral equation, are also discussed and two algorithms are suggested. Numerical experiments are presented to confirm the superconvergence analysis and to show the efficiency of the algorithms.     

Computation of poisson type integrals

We consider problems of computing the Poisson integral when the point at which the integral is evaluated approaches the ball surface. Techniques are proposed enabling one to improve the computation efficiency.     

Mixed boundary integral methods for Helmholtz transmission problems

In this paper we propose a hybrid between direct and indirect boundary integral methods to solve a transmission problem for the Helmholtz equation in Lipschitz and smooth domains. We present an exhaustive abstract study of the numerical approximation of the resulting system of boundary integral equations by means of Galerkin methods. Some particular examples of convergent schemes in the smooth case in two dimensions are given. Finally, we extend the results to a thermal scattering problem in a half plane with several obstacles and provide numerical results that illustrate the accuracy of our methods depending on the regularity of the interface.     

Localization of the Aronszajn-Slobodeckij norm and application to adaptive boundary element methods Part II. The three-dimensional case

Summary. In this paper we introduce new local a-posteriori error indicators for the Galerkin discretization of three-dimensional boundary integral equations. These error indicators are efficient and reliable for a wide class of integral operators, in particular for operators of negative order. They are based on local norms of the computable residual and can be used for controlling the adaptive refinement. The proofs of efficiency and reliability are based on the result that the Aronszajn-Slobodeckij norm (given by a double integral for a non-integer ) is localizable for certain functions. Neither inverse estimates nor saturation properties are needed. In this paper, we extend the two-dimensional results of a previous paper to the three-dimensional case. Received March 20, 2000 / Published online November 15, 2001     

Local a-posteriori error indicators for the Galerkin discretization of boundary integral equations

In this paper we present local a-posteriori error indicators for the Galerkin discretization of boundary integral equations. These error indicators are introduced and investigated by Babuška-Rheinboldt [3] for finite element methods. We transfer them from finite element methods onto boundary element methods and show that they are reliable and efficient for a wide class of integral operators under relatively weak assumptions. These local error indicators are based on the computable residual and can be used for controlling the adaptive mesh refinement. Received March 4, 1996 / Revised version received September 25, 1996     

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.     

Efficient quadrature for highly oscillatory integrals involving critical points

This paper based on the Levin collocation method and Levin-type method together with composite two-point Gauss–Legendre quadrature presents efficient quadrature for integral transformations of highly oscillatory functions with critical points. The effectiveness and accuracy of the quadrature are tested.     

Dirac delta methods for Helmholtz transmission problems

In this paper we use a boundary integral method with single layer potentials to solve a class of Helmholtz transmission problems in the plane. We propose and analyze a novel and very simple quadrature method to solve numerically the equivalent system of integral equations which provides an approximation of the solution of the original problem with linear convergence (quadratic in some special cases). Furthermore, we also investigate a modified quadrature approximation based on the ideas of qualocation methods. This new scheme is again extremely simple to implement and has order three in weak norms.      

A note on an extension of extrapolative techniques for a class of infinite oscillatory integrals

Recent work on extrapolative processes for a class of infinite oscillatory integrals is extended by a procedure which involves the unknown abscissae at which the remainder integral vanishes identically. The asymptotic proximity of these abscissae to a known set is used as a basis for further extrapolation.Results show that the method can be useful for the very slowly converging integrals whose amplitudes decay only algebraically.     

Finite element methods for optimal control problems governed by integral equations and integro-differential equations

In this paper, we analyze finite-element Galerkin discretizations for a class of constrained optimal control problems that are governed by Fredholm integral or integro-differential equations. The analysis focuses on the derivation of a priori error estimates and a posteriori error estimators for the approximation schemes.Grants, communicated-by lines, or other notes about the article will be placed here between rules. Such notes are optional.     

Strictly Singular and Regular Integral Operators

In the setting of K?the function spaces we present some suffcient conditions for a regular integral operator to be strictly singular.     

The construction of some efficient preconditioners in the boundary element method

The discretization of first kind boundary integral equations leads in general to a dense system of linear equations, whose spectral condition number depends on the discretization used. Here we describe a general preconditioning technique based on a boundary integral operator of opposite order. The corresponding spectral equivalence inequalities are independent of the special discretization used, i.e., independent of the triangulations and of the trial functions. Since the proposed preconditioning form involves a (pseudo)inverse operator, one needs for its discretization only a stability condition for obtaining a spectrally equivalent approximation. This revised version was published online in June 2006 with corrections to the Cover Date.     

Quasi-interpolation projectors for box splines

We consider box spline quasi-interpolants based on local linear functionals of point evaluator and integral type. The approximations are easy to compute, and reproduce the whole spline space in question.     

On the numerical solution of a logarithmic integral equation of the first kind for the Helmholtz equation

Summary We describe a quadrature method for the numerical solution of the logarithmic integral equation of the first kind arising from the single-layer approach to the Dirichlet problem for the two-dimensional Helmholtz equation in smooth domains. We develop an error analysis in a Sobolev space setting and prove fast convergence rates for smooth boundary data.     

On summation formulas due to Plana, Lindelöf and Abel, and related Gauss-christoffel rules, I

Three methods, old but not so well known, transform an infinite series into a complex integral over an infinite interval. Gauss quadrature rules are designed for each of them. Various questions concerning their construction and application are studied, theoretically or experimentally. They are so efficient that they should be considered for the development of software for special functions. Applications are made to slowly convergent alternating and positive series, to Fourier series, to the numerical analytic continuation of power series outside the circle of convergence, and to ill-conditioned power series.     

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     

Interior penalty discontinuous approximations of convection–diffusion problems with parabolic layers

Summary A nonsymmetric discontinuous Galerkin finite element method with interior penalties is considered for two–dimensional convection–diffusion problems with regular and parabolic layers. On an anisotropic Shishkin–type mesh with bilinear elements we prove error estimates (uniformly in the perturbation parameter) in an integral norm associated with this method. On different types of interelement edges we derive the values of discontinuity–penalization parameters. Numerical experiments complement the theoretical results.