Numerical methods and asymptotic error expansions for the Emden-Fowler equations

In the present paper we analyse a numerical method for computing the solution of some boundary-value problems for the Emden-Fowler equations. The differential equations are discretized by a finite-difference method and we derive asymptotic expansions for the discretization error. Based on these asymptotic expansions, we use an extrapolation algorithm to accelerate the convergence of the numerical method.     

Asymptotic Error Expansions for Numerical Solutions of Integral Equations

A general theorem dealing with asymptotic error expansions fornumerical solutions of linear operator equations is proved.This is applied to the Nystr?m, collocation, and Galerkin methodsfor second kind, Fredholm integral equations. For example, weshow that when piecewise polynomials of degree m–1 areused, the iterated Galerkin solution admits an error expansionin even powers of the step-size h, beginning with a term inh^2m.     

A boundary integral equation method for three-dimensional crack problems in elasticity

This paper presents a solution procedure for three-dimensional crack problems via first kind boundary integral equations on the crack surface. The Dirichlet (Neumann) problem is reduced to a system of integral equations for the jump of the traction (of the field) across the crack surface. The calculus of pseudodifferential operators is used to derive existence and regularity of the solutions of the integral equations. With the concept of the principal symbol and the Wiener-Hopf technique we derive the explicit behavior of the densities of the integral equations near the edge of the crack surface. Based on the detailed regularity results we show how to improve the boundary element Galerkin method for our integral equations. Quasi-optimal asymptotic estimates for the Galerkin error are given.     

On spline collocation methods for boundary integral equations in the plane

Nowadays boundary elemen; methods belong to the most popular numerical methods for solving elliptic boundary value problems. They consist in the reduction of the problem to equivalent integral equations (or certain generalizations) on the boundary Γ of the given domain and the approximate solution of these boundary equations. For the numerical treatment the boundary surface is decomposed into a finite number of segments and the unknown functions are approximated by corresponding finite elements and usually determined by collocation and Galerkin procedures. One finds the least difficulties in the theoretical foundation of the convergence of Galerkin methods for certain classes of equations, whereas the convergence of collocation methods, which are mostly used in numerical computations, has yet been proved only for special equations and methods. In the present paper we analyse spline collocation methods on uniform meshes with variable collocation points for one-dimensional pseudodifferential equations on a closed curve with convolutional principal parts, which encompass many classes of boundary integral equations in the plane. We give necessary and sufficient conditions for convergence and prove asymptotic error estimates. In particular we generalize some results on nodal and midpoint collocation obtained in [2], [7] and [8]. The paper is organized as follows. In Section 1 we formulate the problems and the results, Section 2 deals with spline interpolation in periodic Sobolev spaces, and in Section 3 we prove the convergence theorems for the considered collocation methods.     

Richardson extrapolation of iterated discrete projection methods for eigenvalue approximation

In this paper, the eigenvalue approximation of a compact integral operator with a smooth kernel is discussed. We propose asymptotic error expansions of the iterated discrete Galerkin and iterated discrete collocation methods, and asymptotic error expansion of approximate eigenvalues. We then apply Richardson extrapolation to obtain higher order super-convergence of eigenvalue approximations. Numerical examples are presented to illustrate the theoretical estimate.     

Defect correction from a Galerkin viewpoint

Summary We consider the numerical solution of systems of nonlinear two point boundary value problems by Galerkin's method. An initial solution is computed with piecewise linear approximating functions and this is then improved by using higher-order piecewise polynomials to compute defect corrections. This technique, including numerical integration, is justified by typical Galerkin arguments and properties of piecewise polynomials rather than the traditional asymptotic error expansions of finite difference methods.     

两种群竞争模型的奇摄动群

本文应用多重尺度法构造出非线性微分方程组的解的渐近展开式。并用微分不等式的技巧，证明原问题的解的存在性，且给出解的一致有效渐近估计．     

An improved boundary element method for the charge density of a thin electrified plate in ℝ3

A weakly singular integral equation of the first kind on a plane surface piece Γ is solved approximately via the Galerkin method. The determination of the solution of this integral equation (with the single-layer potential) is a classical problem in physics, since its solution represents the charge density of a thin, electrified plate Γ loaded with some given potential. Using piecewise constant or piecewise bilinear boundary elements we derive asymptotic estimates for the Galerkin error in the energy norm and analyse the effect of graded meshes. Estimates in lower order Sobolev norms are obtained via the Aubin–Nitsche trick. We describe in detail the numerical implementation of the Galerkin method with both piecewise-constant and piecewise-linear boundary elements. Numerical experiments show experimental rates of convergence that confirm our theoretical, asymptotic results.     

Linearly implicit time discretization of non-linear parabolic equations

We give a stability and error analysis of linearly implicitone-step methods for time discretization of non-linear parabolicequations. We derive precise error bounds for Rosenbrock andW-methods, and we explain the error reduction by Richardsonextrapolation of the linearly implicit Euler method which occursin spite of the breakdown of asymptotic expansions. The parabolicequations are studied in a Hilbert space framework that includessemilinear and quasilinear parabolic equations, and also stiffreaction-diffusion equations with reactions at different timescales.     

Linearly implicit extrapolation methods for differential-algebraic systems

Summary We consider the numerical solution of implicit differential equations in which the solution derivative appears multiplied by a solution-dependent singular matrix. We study extrapolation methods based on two linearly implicit Euler discretizations. Their error behaviour is explained by perturbed asymptotic expansions.     

On a class of singular boundary value problems with singular perturbation

In this paper we investigate a class of singular second order differential equations with singular perturbation subject to three-point boundary value conditions, whose solution exhibits a couple of boundary layers at two endpoints. We first establish a lower–upper solutions theorem by using the Schauder fixed point theorem. By the asymptotic expansions and the lower–upper solutions theorem we obtain the existence, asymptotic estimates and uniqueness for the proposed problem. Several examples are given for illustrating our results.     

Finite element analysis of parabolic integro-differential equations of Kirchhoff type

The aim of this paper is to study parabolic integro-differential equations of Kirchhoff type. We prove the existence and uniqueness of the solution for this problem via Galerkin method. Semidiscrete formulation for this problem is presented using conforming finite element method. As a consequence of the Ritz–Volterra projection, we derive error estimates for both semidiscrete solution and its time derivative. To find the numerical solution of this class of equations, we develop two different types of numerical schemes, which are based on backward Euler–Galerkin method and Crank–Nicolson–Galerkin method. A priori bounds and convergence estimates in spatial as well as temporal direction of the proposed schemes are established. Finally, we conclude this work by implementing some numerical experiments to confirm our theoretical results.     

An improved boundary element Galerkin method for three-dimensional crack problems

In this paper we analyze the solution of crack problems in three-dimensional linear elasticity by equivalent integral equations of the first kind on the crack surface. Besides existence and uniqueness we give sharp regularity results for the solution of these pseudodifferential equations. Two versions of Eskin's Wiener-Hopf technique are presented: the first one requires the factorization of matrix-valued symbols which is avoided in the second case. Based on these regularity results we show how to improve the boundary element Galerkin method for our integral equations by using special singular trial functions. We apply the approximation property and inverse assumption of these elements together with duality arguments and derive quasi-optimal asymptotic error estimates in a scale of Sobolev spaces.Dedicated to Prof. Dr.-Ing. W. L. Wendland on the occasion of his 50th birthday.A part of this work was done while the first author was a guest at the Georgia Institute of Technology and while the second author was partially supported by the NSF grant DMS-8501797.     

Differential equations with degeneration and resurgent analysis

In the present paper, we consider differential equations with cusp-type degeneration in the principal symbol. We prove the existence of an endlessly continuable solution in the case of Fredholm equations with endlessly continuable right-hand side. For the case in which the principal symbol of the operator has simple singularities, we obtain asymptotic expansions of the solutions of equations with degeneration.     

Full collocation methods for some boundary integral equations

In this paper we propose a fully discretized version of the collocation method applied to integral equations of the first kind with logarithmic kernel. After a stability and convergence analysis is given, we prove the existence of an asymptotic expansion of the error, which justifies the use of Richardson extrapolation. We further show how these expansions can be translated to a new expansion of potentials calculated with the numerical solution of a boundary integral equation such as those treated before. Some numerical experiments, confirming our theoretical results, are given. This revised version was published online in August 2006 with corrections to the Cover Date.     

非线性积分方程迭代配置法的渐近展开及其外推

非线性积分方程迭代配置法的渐近展开及其外推韩国强（华南理工大学计算机工程与科学系）ＡＳＹＭＰＴＯＴＩＣＥＲＲＯＲＥＸＭＮＳＩＯＮＳＡＮＤＥＸＴＲＡＰＯＬＡＴＩＯＮＦＯＲＴＨＥＩＴＥＲＡＴＥＤＣＯＬＬＯＣＡＴＩＯＮＭＥＴＨＯＤＳＯＦＮＯＮＬＩＮＥＡＲＩ...     

Defect correction and a posteriori error estimation of Petrov-Galerkin methods for nonlinear Volterra integro-differential equations

We present two defect correction schemes to accelerate the Petrov-Galerkin finite element methods [19] for nonlinear Volterra integro-differential equations. Using asymptotic expansions of the errors, we show that the defect correction schemes can yield higher order approximations to either the exact solution or its derivative. One of these schemes even does not impose any extra regularity requirement on the exact solution. As by-products, all of these higher order numerical methods can also be used to form a posteriori error estimators for accessing actual errors of the Petrov-Galerkin finite element solutions. Numerical examples are also provided to illustrate the theoretical results obtained in this paper.     

具非线性边界条件的半线性时滞微分方程边值问题奇摄动

利用微分不等式理论研究了一类具非线性边界条件的半线性时滞微分方程边值问题.采用新的方法构造上下解,得到了此边值问题解的存在性的充分条件,并给出了解的一致有效渐近展开式.     

二维N-S方程的Fourier非线性Galerkin方法*

本文对周期边界条件Navier-Stokes方程,证明了其Fourier非线性Galerkin逼近解的存在唯一性,同时给出了逼近解的误差估计。     

二维N－S方程的Fourier非线性Galerkin方法

本文对周期边界条件Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ方程，证明了其Ｆｏｕｒｉｅｒ非线性Ｇａｌｅｒｋｉｎ逼近解的存在唯一性，同时给出了逼近解的误差估计．