Splitting extrapolation algorithm for first kind boundary integral equations with singularities by mechanical quadrature methods

The accuracy of numerical solutions near singular points is crucial for numerical methods. In this paper we develop an efficient mechanical quadrature method (MQM) with high accuracy. The following advantages of MQM show that it is very promising and beneficial for practical applications: (1) the O(h_max³) O(h_{\rm {max}}^{3}) convergence rate; (2) the O(h_max⁵)O(h_{\rm {max}}^{5}) convergence rate after splitting extrapolation; (3) Cond = O(h_min^-1)O(h_{\rm {min}}^{-1}); (4) the explicit discrete matrix entries. In this paper, the above theoretical results are briefly addressed and then verified by numerical experiments. The solutions of MQM are more accurate than those of other methods. Note that for the discontinuous model in Li et al. (Eng Anal Bound Elem 29:59–75, 2005), the highly accurate solutions of MQM may even compete with those of the collocation Trefftz method.     

Extrapolation method for solving weakly singular nonlinear Volterra integral equations of the second kind

Based on a new generalization of discrete Gronwall inequality in [L. Tao, H. Yong, A generalization of discrete Gronwall inequality and its application to weakly singular Volterra integral equality of the second kind, J. Math. Anal. Appl. 282 (2003) 56-62], Navot's quadrature rule for computing integrals with the end point singularity in [I. Navot, A further extension of Euler-Maclaurin summation formula, J. Math. Phys. 41 (1962) 155-184] and a transformation in [P. Baratella, A. Palamara Orsi, A new approach to the numerical solution of weakly singular Volterra integral equations, J. Comput. Appl. Math. 163 (2004) 401-418], a new quadrature method for solving nonlinear weakly singular Volterra integral equations of the second kind is presented. The convergence of the approximation solution and the asymptotic expansion of the error are proved, so by means of the extrapolation technique we not only obtain a higher accuracy order of the approximation but also get a posteriori estimate of the error.     

Mechanical quadrature methods and their extrapolation for solving first kind Abel integral equations

This paper presents high accuracy mechanical quadrature methods for solving first kind Abel integral equations. To avoid the ill-posedness of problem, the first kind Abel integral equation is transformed to the second kind Volterra integral equation with a continuous kernel and a smooth right-hand side term expressed by weakly singular integrals. By using periodization method and modified trapezoidal integration rule, not only high accuracy approximation of the kernel and the right-hand side term can be easily computed, but also two quadrature algorithms for solving first kind Abel integral equations are proposed, which have the high accuracy O(h²)$\mathrm{O}(h^2)$ and asymptotic expansion of the errors. Then by means of Richardson extrapolation, an approximation with higher accuracy order O(h³)$\mathrm{O}(h^3)$ is obtained. Moreover, an a posteriori error estimate for the algorithms is derived. Some numerical results show the efficiency of our methods.     

Block boundary value methods for solving Volterra integral and integro-differential equations

Reducible quadrature rules generated by boundary value methods are considered in block version and applied to solve the second kind Volterra integral equations and Volterra integro-differential equations. These extended block boundary value methods are shown to possess both excellent stability properties and high accuracy for Volterra-type equations. Numerical experiments are presented and the efficiency, accuracy and stability of the schemes are confirmed.     

Numerical methods of solving nonlinear integral equations of volterra type

Applying continued fractions. we propose numerical methods of solving nonlinear Volterra integral equations of second kind. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 39, No. 2, 1996, pp. 127–132     

Product integration methods for solving a system of nonlinear Volterra integral equations

In this paper the technique of subtracting out singularities is used to derive explicit and implicit product Euler schemes with order one convergence and a product trapezoidal scheme with order two convergence for a system of Volterra integral equations with a weakly singular kernel. The convergence proofs of the numerical schemes are presented; these are nonstandard since the nonlinear function involved in the integral equation system does not satisfy a global Lipschitz condition.     

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.     

Iterative methods for solving fredholm integral equations

A Gauss-Seidel type of iterative method is described for solving the non-linear Fredholm integral equation. The analysis shows that this method may be expected to converge faster than the standard iterative method.     

A quadrature method for solving integral equations of the mixed type

This paper is dedicated to quadrature methods for solving integro-operational equations. We construct computational schemes of quadrature methods and substantiate them theoretically in the sense of the general theory of approximate methods developed by L. V. Kantorovich and B. G. Gabdulkhaev.     

Extrapolation methods to compute hypersingular integral in boundary element methods

The composite trapezoidal rule for the computation of Hadamard finite-part integrals in boundary element methods with the hypersingular kernel 1/sin 2(x-s) is discussed,and the main part of the asymptotic expansion of error function is obtained.Based on the main part of the asymptotic expansion,a series is constructed to approach the singular point.An extrapolation algorithm is presented and the convergence rate is proved.Some numerical results are also presented to confirm the theoretical results and show the efficiency of the algorithms.     

Multi-parameter extrapolation methods for boundary integral equations

Multi-parameter extrapolation was first introduced by Zhou et al. for solving partial differential equations with finite element methods in 1994. The method is based on a domain decomposition and independent discretization of the subdomains resulting in a multi-parameter error expansion. This permits a generalized extrapolation technique. The algorithm is naturally parallel since the main computational work is spent in solving independent linear systems. Here the method is extended to the case of boundary integral equations on polygonal domains, where singularities require graded meshes. A complete analysis is given, based on weighted norm techniques. Several numerical experiments demonstrate the mathematical features and practical usefulness of the method. This revised version was published online in June 2006 with corrections to the Cover Date.     

Qualocation methods for elliptic boundary integral equations

Summary. The qualocation methods developed in this paper, with spline trial and test spaces, are suitable for classes of boundary integral equations with convolutional principal part, on smooth closed curves in the plane. Some of the methods are suitable for all strongly elliptic equations; that is, for equations in which the even symbol part of the operator dominates. Other methods are suitable when the odd part dominates. Received December 27, 1996 / Revised version received April 14, 1997     

Fast Fourier-Galerkin methods for solving singular boundary integral equations: Numerical integration and precondition

We develop a fast fully discrete Fourier-Galerkin method for solving a class of singular boundary integral equations. We prove that the number of multiplications used in generating the compressed matrix is O(nlog³n), and the solution of the proposed method preserves the optimal convergence order O(n^−t), where n is the order of the Fourier basis functions used in the method and t denotes the degree of regularity of the exact solution. Moreover, we propose a preconditioning which ensures the numerical stability when solving the preconditioned linear system. Numerical examples are presented to confirm the theoretical estimates and to demonstrate the approximation accuracy and computational efficiency of the proposed algorithm.     

New predictor-corrector methods for solving nonlinear equations

In this paper, we present a new one-step iterative method for solving nonlinear equations, which inherits the advantages of both Newton’s and Steffensen’s methods. Moreover, two two-step methods of second-order are proposed by combining it with the regula falsi method. These new two-step methods present attractive features such as being independent of the initial values in the iterative interval, or being adaptive for the iterative formulas. The convergence of the iterative sequences is deduced. Finally, numerical experiments verify their merits.     

Combined bracketing methods for solving nonlinear equations

Several methods based on combinations of bisection, regula falsi, and parabolic interpolation has been developed. An interval bracketing ensures the global convergence while the combination with the parabolic interpolation increases the speed of the convergence. The proposed methods have been tested on a series of examples published in the literature and show good results.     

Steffensen type methods for solving nonlinear equations

In the present paper, by approximating the derivatives in the well known fourth-order Ostrowski’s method and in a sixth-order improved Ostrowski’s method by central-difference quotients, we obtain new modifications of these methods free from derivatives. We prove the important fact that the methods obtained preserve their convergence orders 4 and 6, respectively, without calculating any derivatives. Finally, numerical tests confirm the theoretical results and allow us to compare these variants with the corresponding methods that make use of derivatives and with the classical Newton’s method.