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...     

A fourth/second order accurate collocation method for singularly perturbed two-point boundary value problems using tension splines

Summary.   The collocation tension spline is considered as a numerical solution of a singularly perturbed two-point boundary value problem: . The collocation points are chosen as a generalization of the classical Gaussian points. Unlike the traditional approach, we employ the B-spline representation in the analysis. This leads to global quadratic convergence of the method for small perturbation parameters, and, for large values, the order of convergence is four. Received October 4, 1996 / Revised version received September 23, 1999 / Published online October 16, 2000     

An optimal two-step quadratic spline collocation method for the Dirichlet biharmonic problem

Numerical Algorithms - A two-step quadratic spline collocation method is formulated for the solution of the Dirichlet biharmonic problem on the unit square rewritten as a coupled system of two...     

Unfitted finite element for optimal control problem of the temperature in composite media with contact resistance

Numerical Algorithms - A new method based on quadratic spline collocation is formulated for the solution of the Dirichlet biharmonic problem on the unit square rewritten as a coupled system of two...     

On a spline collocation method for a singularly perturbed problem

We consider a singularly perturbed boundary value problem with two small parameters. The problem is numerically treated by a quadratic spline collocation method. The suitable choice of collocation points provides the inverse monotonicity enabling utilization of barrier function method in the error analysis. Numerical results give justification of the proposed method. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

二阶常微分方程张力样条配置解的渐近式及其外推算法

首先考虑二阶常微分方程第一边值问题:假设(1)有唯一解,且解u(x)∈C~6[a,b];f(x,y,z)作为x,y,z的函数属于 C~2,     

Multigrid and multilevel methods for quadratic spline collocation

Multigrid methods are developed and analyzed for quadratic spline collocation equations arising from the discretization of one-dimensional second-order differential equations. The rate of convergence of the two-grid method integrated with a damped Richardson relaxation scheme as smoother is shown to be faster than 1/2, independently of the step-size. The additive multilevel versions of the algorithms are also analyzed. The development of quadratic spline collocation multigrid methods is extended to two-dimensional elliptic partial differential equations. Multigrid methods for quadratic spline collocation methods are not straightforward: because the basis functions used with quadratic spline collocation are not nodal basis functions, the design of efficient restriction and extension operators is nontrivial. Experimental results, with V-cycle and full multigrid, indicate that suitably chosen multigrid iteration is a very efficient solver for the quadratic spline collocation equations. Supported by Communications and Information Technology Ontario (CITO), Canada. Supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Computational and Technology Research, U.S. Department of Energy, under Contract W-31-109-Eng-38.     

Numerical solution of differential-algebraic equations using the spline collocation-variation method

Numerical methods for solving initial value problems for differential-algebraic equations are proposed. The approximate solution is represented as a continuous vector spline whose coefficients are found using the collocation conditions stated for a subgrid with the number of collocation points less than the degree of the spline and the minimality condition for the norm of this spline in the corresponding spaces. Numerical results for some model problems are presented.     

Regularized collocation method for Fredholm integral equations of the first kind

In this paper we consider a collocation method for solving Fredholm integral equations of the first kind, which is known to be an ill-posed problem. An “unregularized” use of this method can give reliable results in the case when the rate at which smallest singular values of the collocation matrices decrease is known a priori. In this case the number of collocation points plays the role of a regularization parameter. If the a priori information mentioned above is not available, then a combination of collocation with Tikhonov regularization can be the method of choice. We analyze such regularized collocation in a rather general setting, when a solution smoothness is given as a source condition with an operator monotone index function. This setting covers all types of smoothness studied so far in the theory of Tikhonov regularization. One more issue discussed in this paper is an a posteriori choice of the regularization parameter, which allows us to reach an optimal order of accuracy for deterministic noise model without any knowledge of solution smoothness.     

Optimal superconvergent one step quadratic spline collocation methods

We formulate new optimal quadratic spline collocation methods for the solution of various elliptic boundary value problems in the unit square. These methods are constructed so that the collocation equations can be solved using a matrix decomposition algorithm. The results of numerical experiments exhibit the expected optimal global accuracy as well as superconvergence phenomena. AMS subject classification (2000)  65N35, 65N22     

Spline Collocation Approximation to Periodic Solutions of Ordinary Differential Equations

A spline collocation method is proposed to approximate the periodic solution of nonlinear ordinary differential equations. It is proved that the cubic periodic spline collocation solution has the same error bound $O(h^4)$ and superconvergence of the derivative at collocation points as that of the interpolating spline function. Finally a numerical example is given to demonstrate the effectiveness of our algorithm.     

On the spline collocation method for the single layer equation related to time-fractional diffusion

The boundary element spline collocation method is studied for the time-fractional diffusion equation in a bounded two-dimensional domain. We represent the solution as the single layer potential which leads to a Volterra integral equation of the first kind. We discretize the boundary integral equation with the spline collocation method on uniform meshes both in spatial and time variables. In the stability analysis we utilize the Fourier analysis technique developed for anisotropic pseudodifferential equations. We prove that the collocation solution is quasi-optimal under some stability condition for the mesh parameters. We have to assume that the mesh parameter in time satisfies (h_t=c h^\frac2a)(h_t=c h^{\frac{2}{\alpha}}), where (h) is the spatial mesh parameter.     

A conjugate gradient method for the unconstrained minimization of strictly convex quadratic splines

In this paper, we show that an analogue of the classical conjugate gradient method converges linearly when applied to solving the problem of unconstrained minimization of a strictly convex quadratic spline. Since a strictly convex quadratic program with simple bound constraints can be reformulated as unconstrained minimization of a strictly convex quadratic spline, the conjugate gradient method is used to solve the unconstrained reformulation and find the solution of the original quadratic program. In particular, if the solution of the original quadratic program is nondegenerate, then the conjugate gradient method finds the solution in a finite number of iterations. This author's research is partially supported by the NASA/Langley Research Center under grant NCC-1-68 Supplement-15.     

On the Characterization of Quadratic Splines

A quadratic spline is a differentiable piecewise quadratic function. Many problems in the numerical analysis and optimization literature can be reformulated as unconstrained minimizations of quadratic splines. However, only special cases of quadratic splines have been studied in the existing literature and algorithms have been developed on a case-by-case basis. There lacks an analytical representation of a general or even convex quadratic spline. The current paper fills this gap by providing an analytical representation of a general quadratic spline. Furthermore, for a convex quadratic spline, it is shown that the representation can be refined in the neighborhood of a nondegenerate point and a set of nondegenerate minimizers. Based on these characterizations, many existing algorithms for specific convex quadratic splines are also finitely convergent for a general convex quadratic spline. Finally, we study the relationship between the convexity of a quadratic spline function and the monotonicity of the corresponding linear complementarity problem. It is shown that, although both conditions lead to easy solvability of the problem, they are different in general.This project was initiated when the first author was visiting the Technical University of Denmark and Erasmus University. The visit was partially funded by the Danish Natural Science Research Council.     

Using Richardson's method to construct high-order accurate adaptive grids

An algorithm is proposed to construct a sequence of evolutionary adaptive grids, each consisting of segments of uniform grids compatible with the Richardson extrapolation procedure. The necessary numerical accuracy is achieved by using Richardson's extrapolation method to increase the accuracy of the difference solution and by controlling the nonhomogeneity parameter on passing from one segment to the next. The numerical model used in the article is the diffusion-convection equation, whose solution contains large gradients. Numerical calculations show that the algorithm attains the prespecified accuracy on a nonuniform difference grid. Numerical examples support the universality of the proposed algorithm. High accuracy results can be obtained without changing the structure of the difference schemes approximating the original problem; only the grid spacing has to be changed. Translated from Chislennye Metody v Matematicheskoi Fizike, Moscow State University, pp. 22–36, 1998.     

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.     

Nonconforming spline collocation methods in irregular domains

This article studies a class of nonconforming spline collocation methods for solving elliptic PDEs in an irregular region with either triangular or quadrilateral partition. In the methods, classical Gaussian points are used as matching points and the special quadrature points in a triangle or quadrilateral element are used as collocation points. The solution and its normal derivative are imposed to be continuous at the marching points. The authors present theoretically the existence and uniqueness of the numerical solution as well as the optimal error estimate in H¹‐norm for a spline collocation method with rectangular elements. Numerical results confirm the theoretical analysis and illustrate the high‐order accuracy and some superconvergence features of methods. Finally the authors apply the methods for solving two physical problems in compressible flow and linear elasticity, respectively. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

B‐spline spectral method for constrained fractional optimal control problems

In this paper, we present a direct B‐spline spectral collocation method to approximate the solutions of fractional optimal control problems with inequality constraints. We use the location of the maximum of B‐spline functions as collocation points, which leads to sparse and nonsingular matrix B whose entries are the values of B‐spline functions at the collocation points. In this method, both the control and Caputo fractional derivative of the state are approximated by B‐spline functions. The fractional integral of these functions is computed by the Cox‐de Boor recursion formula. The convergence of the method is investigated. Several numerical examples are considered to indicate the efficiency of the method.     

Spline collocation methods for linear multi-term fractional differential equations

Some regularity properties of the solution of linear multi-term fractional differential equations are derived. Based on these properties, the numerical solution of such equations by piecewise polynomial collocation methods is discussed. The results obtained in this paper extend the results of Pedas and Tamme (2011) [15] where we have assumed that in the fractional differential equation the order of the highest derivative of the unknown function is an integer. In the present paper, we study the attainable order of convergence of spline collocation methods for solving general linear fractional differential equations using Caputo form of the fractional derivatives and show how the convergence rate depends on the choice of the grid and collocation points. Theoretical results are verified by some numerical examples.     

Spline collocation methods for linear multi-term fractional differential equations

Some regularity properties of the solution of linear multi-term fractional differential equations are derived. Based on these properties, the numerical solution of such equations by piecewise polynomial collocation methods is discussed. The results obtained in this paper extend the results of Pedas and Tamme (2011) [15] where we have assumed that in the fractional differential equation the order of the highest derivative of the unknown function is an integer. In the present paper, we study the attainable order of convergence of spline collocation methods for solving general linear fractional differential equations using Caputo form of the fractional derivatives and show how the convergence rate depends on the choice of the grid and collocation points. Theoretical results are verified by some numerical examples.