Error estimates for shooting methods in two-point boundary value problems for second-order equations

This paper is concerned with a procedure for estimating the global discretization error arising when a boundary value problem for a system of second order differential equations is solved by the simple shooting method, without transforming the original problem in an equivalent first order problem. Expressions of the global discretization error are derived for both linear and nonlinear boundary value problems, which reduce the error estimation for a boundary value problem to that for an initial value problem of same dimension. The procedure extends to second order equations a technique for global error estimation given elsewhere for first order equations. As a practical result the accuracy of the estimates for a second order problem is increased compared with the estimates for the equivalent first order problem.     

基于最佳超收敛阶EEP法的一维有限元自适应求解

基于新近提出的具有最佳超收敛阶的单元能量投影（EEP）超收敛算法,提出用具有最佳超收敛阶的EEP超收敛解对有限元解进行误差估计,用均差法进行网格划分,用拟有限元解进行多次遍历而不反复求解有限元真解,形成一套新型的一维有限元自适应求解策略．该法理论上简明清晰,算法上高效可靠,对于大多数问题,一步自适应迭代便可给出按最大模度量逐点满足误差限的有限元解答．以二阶椭圆型常微分方程模型问题为例,介绍了该法的基本思想、实施策略及具体算法,并给出具有代表性的数值算例,以展示该法的优良性能和效果．     

Multigrid method for coupled semilinear elliptic equation

This paper introduces a kind of multigrid finite element method for the coupled semilinear elliptic equations. Instead of the common way of directly solving the coupled semilinear elliptic problems on some fine spaces, the presented method transforms the solution of the coupled semilinear elliptic problem into a series of solutions of the corresponding decoupled linear boundary value problems on the sequence of multilevel finite element spaces and some coupled semilinear elliptic problems on a very low dimensional space. The decoupled linearized boundary value problems can be solved by some multigrid iterations efficiently. The optimal error estimate and optimal computational work are proved theoretically and demonstrated numerically. Moreover, the requirement of bounded second‐order derivatives of the nonlinear term in the existing multigrid method is reduced to a Lipschitz continuous condition in the proposed method.     

Optimal recovery of integral operators and its applications

In this paper we present the solution to a problem of recovering a rather arbitrary integral operator based on incomplete information with error. We apply the main result to obtain optimal methods of recovery and compute the optimal error for the solutions to certain integral equations as well as boundary and initial value problems for various PDE’s.     

An application of the exponential cubic splines to numerical solution of a self-adjoint perturbation problem

We present a class of the second order optimal splines difference schemes derived from expomential cubic splines for self-adjoint singularly perturbed 2-point boundary value problem. We prove an optimal error estimate and give illustrative numerical example.     

OPTIMAL ERROR ESTIMATES OF THE PARTITION OF UNITY METHOD WITH LOCAL POLYNOMIAL APPROXIMATION SPACES

In this paper, we provide a theoretical analysis of the partition of unity finite elementmethod (PUFEM), which belongs to the family of meshfree methods. The usual erroranalysis only shows the order of error estimate to the same as the local approximations[12].Using standard linear finite element base functions as partition of unity and polynomials aslocal approximation space, in 1-d case, we derive optimal order error estimates for PUFEMinterpolants. Our analysis show that the error estimate is of one order higher than thelocal approximations. The interpolation error estimates yield optimal error estimates forPUFEM solutions of elliptic boundary value problems.     

$L^\infty$ Convergence of TRUNC Element for the Biharmonic Equation

The paper considers the $L^\infty$ convergence for TRUNC finite elements solving the boundary value problems of the biharmonic equation. The nearly optimal $L^\infty$ estimates for the error of first order derivatives are given.     

Raviart-Thomas混合元的超收敛

考虑二阶椭圆方程Dirichlet边值问题在正则矩形网格上k阶RaviartThomas混合有限元的超收敛.对有限元解经插值处理后,与通常的有限元最优误差估计相比,收敛速度提高了两阶.     

Asymptotic properties of solutions of parametric optimal control problems with varying index of the singular arcs

We consider a family of parametric linear-quadratic optimal control problems with terminal and control constraints. This family has the specific feature that the class of optimal controls is changed for an arbitrarily small change in the parameter. In the perturbed problem, the behavior of the corresponding trajectory on noncritical arcs of the optimal control is described by solutions of singularly perturbed boundary value problems. For the solutions of these boundary value problems, we obtain an asymptotic expansion in powers of the small parameter ?. The asymptotic formula starts from a term of the order of 1/? and contains boundary layers. This formula is used to justify the asymptotic expansion of the optimal control for a perturbed problem in the family. We suggest a simple method for constructing approximate solutions of the perturbed optimal control problems without integrating singularly perturbed systems. The results of a numerical experiment are presented.     

The role of conditioning in mesh selection algorithms for first order systems of linear two point boundary value problems

Codes for the numerical solution of two-point boundary value problems can now handle quite general problems in a fairly routine and reliable manner. When faced with particularly challenging equations, such as singular perturbation problems, the most efficient codes use a highly non-uniform grid in order to resolve the non-smooth parts of the solution trajectory. This grid is usually constructed using either a pointwise local error estimate defined at the grid points or else by using a local residual control. Similar error estimates are used to decide whether or not to accept a solution. Such an approach is very effective in general providing that the problem to be solved is well conditioned. However, if the problem is ill conditioned then such grid refinement algorithms may be inefficient because many iterations may be required to reach a suitable mesh on which to compute the solution. Even worse, for ill conditioned problems an inaccurate solution may be accepted even though the local error estimates may be perfectly satisfactory in that they are less than a prescribed tolerance. The primary reason for this is, of course, that for ill conditioned problems a small local error at each grid point may not produce a correspondingly small global error in the solution. In view of this it could be argued that, when solving a two-point boundary value problem in cases where we have no idea of its conditioning, we should provide an estimate of the condition number of the problem as well as the numerical solution. In this paper we consider some algorithms for estimating the condition number of boundary value problems and show how this estimate can be used in the grid refinement algorithm.     

带松弛因子的Schwarz交替方法

§1.引言 Schwarz交替方法的收敛速度,依赖于子区域重迭部分的大小,重迭部分越大,收敛越快。然而重迭部分增大,必将引起计算量的增大,因此,在重迭部分不变的情况下,如何改善Schwarz交替过程的收敛速度,已成为人们感兴趣的问题。关于Schwarz算法收敛速度的讨论,许多文章都是对具体类型的微分方程展开的。     

Lavrentiev regularization + Ritz approximation = uniform finite element error estimates for differential equations with rough coefficients

We consider a parametric family of boundary value problems for a diffusion equation with a diffusion coefficient equal to a small constant in a subdomain. Such problems are not uniformly well-posed when the constant gets small. However, in a series of papers, Bakhvalov and Knyazev have suggested a natural splitting of the problem into two well-posed problems. Using this idea, we prove a uniform finite element error estimate for our model problem in the standard parameter-independent Sobolev norm. We also study uniform regularity of the transmission problem, needed for approximation. A traditional finite element method with only one additional assumption, namely, that the boundary of the subdomain with the small coefficient does not cut any finite element, is considered.

One interpretation of our main theorem is in terms of regularization. Our FEM problem can be viewed as resulting from a Lavrentiev regularization and a Ritz-Galerkin approximation of a symmetric ill-posed problem. Our error estimate can then be used to find an optimal regularization parameter together with the optimal dimension of the approximation subspace.

     

HIGH ACCURACY FINITE VOLUME ELEMENT METHOD FOR TWO-POINT BOUNDARY VALUE PROBLEM OF SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS

1　ＩｎｔｒｏｄｕｃｔｉｏｎＦｉｎｉｔｅｖｏｌｕｍｅｅｌｅｍｅｎｔｍｅｔｈｏｄｏｒＦＶＥＭｕｓｅｓａｖｏｌｕｍｅｉｎｔｅｇｒａｌｆｏｒｍｕｌａｔｉｏｎｏｆｔｈｅｄｉｆ ｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｗｉｔｈａｆｉｎｉｔｅｐａｒｔｉｔｉｏｎｉｎｇｓｅｔｏｆｖｏｌｕｍｅｔｏｄｉｓｃｒｅｔｉｚｅｔｈｅｅｑｕａｔｉｏｎ ,ｔｈｅｎｒｅ ｓｔｒｉｃｔｓｔｈｅａｄｍｉｓｓｉｂｌｅｆｕｎｃｔｉｏｎｓｔｏａｌｉｎｅａｒｆｉｎｉｔｅｅｌｅｍｅｎｔｓｐａｃｅｔｏｄｉｓｃｒｅｔｉｚｅｔｈｅｓｏｌｕｔｉｏｎ ([1 -4 ] ) .…     

Meshless Galerkin algorithms for boundary integral equations with moving least square approximations

In this paper, we first give error estimates for the moving least square (MLS) approximation in the Hk norm in two dimensions when nodes and weight functions satisfy certain conditions. This two-dimensional error results can be applied to the surface of a three-dimensional domain. Then combining boundary integral equations (BIEs) and the MLS approximation, a meshless Galerkin algorithm, the Galerkin boundary node method (GBNM), is presented. The optimal asymptotic error estimates of the GBNM for three-dimensional BIEs are derived. Finally, taking the Dirichlet problem of Laplace equation as an example, we set up a framework for error estimates of the GBNM for boundary value problems in three dimensions.     

B-spline collocation for solution of two-point boundary value problems

A numerical method based on B-spline is developed to solve the general nonlinear two-point boundary value problems up to order 6. The standard formulation of sextic spline for the solution of boundary value problems leads to non-optimal approximations. In order to derive higher orders of accuracy, high order perturbations of the problem are generated and applied to construct the numerical algorithm. The error analysis and convergence properties of the method are studied via Green’s function approach. O(h⁶) global error estimates are obtained for numerical solution of these classes of problems. Numerical results are given to illustrate the efficiency of the proposed method. Results of numerical experiments verify the theoretical behavior of the orders of convergence.     

A new method for solving the nonlinear second-order boundary value differential equations

In this paper we use measure theory to solve a wide range of second-order boundary value ordinary differential equations. First, we transform the problem to a first order system of ordinary differential equations (ODE’s) and then define an optimization problem related to it. The new problem is modified into one consisting of the minimization of a linear functional over a set of Radon measures; the optimal measure is then approximated by a finite combination of atomic measures and the problem converted approximatly to a finite-dimensional linear programming problem. The solution to this problem is used to construct the approximate solution of the original problem. Finally we get the error functionalE (we define in this paper) for the approximate solution of the ODE’s problems.     

Runge-Kutta methods without order reduction for linear initial boundary value problems

Summary. It is well-known the loss of accuracy when a Runge–Kutta method is used together with the method of lines for the full discretization of an initial boundary value problem. We show that this phenomenon, called order reduction, is caused by wrong boundary values in intermediate stages. With a right choice, the order reduction can be avoided and the optimal order of convergence in time is achieved. We prove this fact for time discretizations of abstract initial boundary value problems based on implicit Runge–Kutta methods. Moreover, we apply these results to the full discretization of parabolic problems by means of Galerkin finite element techniques. We present some numerical examples in order to confirm that the optimal order is actually achieved. Received July 10, 2000 / Revised version received March 13, 2001 / Published online October 17, 2001     

Superconvergence analysis and error expansion for the Wilson nonconforming finite element

Summary. In this paper the Wilson nonconforming finite element is considered for solving a class of two-dimensional second-order elliptic boundary value problems. Superconvergence estimates and error expansions are obtained for both uniform and non-uniform rectangular meshes. A new lower bound of the error shows that the usual error estimates are optimal. Finally a discussion on the error behaviour in negative norms shows that there is generally no improvement in the order by going to weaker norms. Received July 5, 1993     

An asymptotic initial value method for second order singular perturbation problems of convection-diffusion type with a discontinuous source term

In this paper a numerical method is presented to solve singularly perturbed two points boundary value problems for second order ordinary differential equations consisting a discontinuous source term. First, in this method, an asymptotic expansion approximation of the solution of the boundary value problem is constructed using the basic ideas of a well known perturbation method WKB. Then some initial value problems and terminal value problems are constructed such that their solutions are the terms of this asymptotic expansion. These initial value problems are happened to be singularly perturbed problems and therefore fitted mesh method (Shishkin mesh) are used to solve these problems. Necessary error estimates are derived and examples provided to illustrate the method.     

Goal-oriented error estimates including algebraic errors in discontinuous Galerkin discretizations of linear boundary value problems

We deal with a posteriori error control of discontinuous Galerkin approximations for linear boundary value problems. The computational error is estimated in the framework of the Dual Weighted Residual method (DWR) for goal-oriented error estimation which requires to solve an additional (adjoint) problem. We focus on the control of the algebraic errors arising from iterative solutions of algebraic systems corresponding to both the primal and adjoint problems. Moreover, we present two different reconstruction techniques allowing an efficient evaluation of the error estimators. Finally, we propose a complex algorithm which controls discretization and algebraic errors and drives the adaptation of the mesh in the close to optimal manner with respect to the given quantity of interest.