AnO(h 6) quintic spline collocation method for fourth order two-point boundary value problems

AnO(h ⁶) collocation method based on quintic splines is developed and analyzed for general fourth-order linear two-point boundary value problems. The method determines a quintic spline approximation to the solution by forcing it to satisfy a high order perturbation of the original boundary value problem at the nodal points of the spline. A variation of this method is formulated as a deferred correction method. The error analysis of the new method and its numerical behavior is presented.This research was supported by AFOSR grant 84-0385.     

On the approximation of nonlinear singular self-adjoint second order boundary value problems

We investigate the approximation of the solutions of a class of nonlinear second order singular boundary value problems with a self-adjoint linear part. Our strategy involves two ingredients. First, we take advantage of certain boundary condition functions to obtain well behaved functions of the solutions. Second, we integrate the problem over an interval that avoids the singularity. We are able to prove a uniform convergence result for the approximate solutions. We describe how the approximation is constructed for the various values of the deficiency index associated with the differential equation. The solution of the nonlinear problem is obtained by a globally convergent iterative method.     

Extremal solutions of sixth-order boundary value problems

For sixth-order boundary value problems, we find extremal solutions that provide the best estimates in the proof of the existence of a solution by the method of a priori estimates.     

Convergence analysis for least-squares finite element approximations of second-order two-point boundary value problems

In this paper, a C⁰$C^0$ least-squares finite element method for second-order two-point boundary value problems is considered. The problem is recast as a first-order system. Standard and improved optimal error estimates in maximum-norms are established. Superconvergence estimates at interelement, Lobatto, and Gauss points are developed. Numerical experiments are given to illustrate theoretical results.     

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.     

Qualocation for a singularly perturbed boundary value problem

A singularly perturbed one-dimensional two point boundary value problem of reaction–convection–diffusion type is considered. We generate a C⁰$C^0$-collocation-like method by combining Galerkin with an adapted quadrature rule. Using Lobatto quadrature and splines of degree r$r$, we prove on a Shishkin mesh for the qualocation method the same error estimate as for the Galerkin technique. The result is also important for the practical realization of finite element methods on Shishkin meshes using quadrature formulas. We report the results of numerical experiments that support the theoretical findings.     

L p estimates of boundary integral equations for some nonlinear boundary value problems

Summary Recently, Galerkin and collocation methods have been analyzed for boundary integral equation formulations of some potential problems in the plane with nonlinear boundary conditions, and stability results and error estimates in theH ^1/2-norm have been proved (Ruotsalainen and Wendland, and Ruotsalainen and Saranen). We show that these results extend toL ^p setting without any extra conditions. These extensions are proved by studying the uniform boundedness of the inverses of the linearized integral operators, and then considering the nonlinear equations. The fact that inH ^1/2 setting the nonlinear operator is a homeomorphism with Lipschitz continuous inverse plays a crucial role. Optimal error estimates for the Galerkin and collocation method inL ^p space then follow.This research was performed while the second author was visiting professor at the University of Delaware, spring 1989     

Symmetric collocation methods for linear differential-algebraic boundary value problems

Summary. We present symmetric collocation methods for linear differential-algebraic boundary value problems without restrictions on the index or the structure of the differential-algebraic equation. In particular, we do not require a separation into differential and algebraic solution components. Instead, we use the splitting into differential and algebraic equations (which arises naturally by index reduction techniques) and apply Gau?-type (for the differential part) and Lobatto-type (for the algebraic part) collocation schemes to obtain a symmetric method which guarantees consistent approximations at the mesh points. Under standard assumptions, we show solvability and stability of the discrete problem and determine its order of convergence. Moreover, we show superconvergence when using the combination of Gau? and Lobatto schemes and discuss the application of interpolation to reduce the number of function evaluations. Finally, we present some numerical comparisons to show the reliability and efficiency of the new methods. Received September 22, 2000 / Revised version received February 7, 2001 / Published online August 17, 2001     

Collocation methods for solving linear differential-algebraic boundary value problems

Summary. We consider boundary value problems for linear differential-algebraic equations with variable coefficients with no restriction on the index. A well-known regularisation procedure yields an equivalent index one problem with d differential and a=n-d algebraic equations. Collocation methods based on the regularised BVP approximate the solution x by a continuous piecewise polynomial of degree k and deliver, in particular, consistent approximations at mesh points by using the Radau schemes. Under weak assumptions, the collocation problems are uniquely and stably solvable and, if the unique solution x is sufficiently smooth, convergence of order min {k+1,2k-1} and superconvergence at mesh points of order 2k-1 is shown. Finally, some numerical experiments illustrating these results are presented. Received October 1, 1999 / Revised version received April 25, 2000 / Published online December 19, 2000     

Orderh 2 method for a singular two-point boundary value problem

A method is described based on auniform mesh for the singular two-point boundary value problem:y+(/x)y+f(x, y)=0, 0<x1,y(0)=0,y(1)=A, and it is shown to be orderh ² convergent forall 1.     

Galerkin-wavelet methods for two-point boundary value problems

Summary Anti-derivatives of wavelets are used for the numerical solution of differential equations. Optimal error estimates are obtained in the applications to two-point boundary value problems of second order. The orthogonal property of the wavelets is used to construct efficient iterative methods for the solution of the resultant linear algebraic systems. Numerical examples are given.This work was supported by National Science Foundation     

Non polynomial splines and weakly singular two-point boundary value problems

In the present paper we shall consider an application of simple non-polynomial splines to a numerical solution of a weakly singular two-point boundary value problem:x ^– (x y)=f(x,y), (0<x1) subject toy(0)=0,y(1)=c ₁(1) ory(0)=c ₂,y(1)=c ₃(0<<1). Our collocation method gives a continuously differentiable approximation and isO(h ²)-convergent.     

Feedback and adaptive finite element solution of one-dimensional boundary value problems

Summary This paper examines the concepts of feedback and adaptivity for the Finite Element Method. The model problem concernsC ⁰ elements of arbitrary, fixed degree for a one-dimensional two-point boundary value problem. Three different feedback methods are introduced and a detailed analysis of their adaptivity is given.Dedicated to F.L. Bauer on the occasion of his 60th birthdayThis research was partially supported by the Office of Naval Research under grant number N00014-77-C-0623     

Stability of some boundary value methods for the solution of initial value problems

The stability properties of three particular boundary value methods (BVMs) for the solution of initial value problems are considered. Our attention is focused on the BVMs based on the midpoint rule, on the Simpson method and on an Adams method of order 3. We investigate their BV-stability regions by considering the scalar test problem and constant stepsize. The study of the conditioning of the coefficient matrix of the discrete problem is extended to the case of variable stepsize and block ODE problems. We also analyse an appropriate choice for the stepsize for stiff problems. Numerical tests are reported to evidentiate the effectiveness of the BVMs and the differences among the BVMs considered.Work supported by the Ministero della Ricerca Scientifica, 40% project, and C.N.R. (contract of research # 92.00535.01).     

On some difference schemes for singular singularly-perturbed boundary value problems

Summary Singularly perturbed boundary value ordinary differential problems are considered, where the problem defining the reduced solution is singular. For numerical approximation, families of symmetric difference schemes, which are equivalent to certain collocation schemes based on Gauss and Lobatto points, are used. Convergence results, previously obtained for the regular singularly perturbed case, are extended. While Gauss schemes are extended with no change, Lobatto schemes require a small modification in the mesh selection procedure. With meshes as prescribed in the text, highly accurate solutions can be obtained with these schemes for singular singularly perturbed problems at a very reasonable cost. This is demonstrated by examples.This research was completed while the author was visiting the Department of Applied Mathematics, Weizmann Inst., Rehovot, Israel. The author was supported in part under NSERC grant A4306