Higher order methods for a singularly perturbed problem

Summary A finite element method using piecewise polynomials of degree k is used to approximate the problem u+u=f, >0 a small parameter. A very irregular mesh is used. On this mesh error estimates of order0(h ^k+1) are obtained uniformly in ,h the maximum stepsize, fork2. The condition number of the system of linear equations one has to solve in order to get the approximation is estimated. Extension of the results to more complicated problems is briefly indicated. Finally, a numerical example is given.Work performed while visiting the IBM Thomas J. Watson Research Center, Yorktown Heights, N.Y.     

A uniformly convergent difference scheme for a semilinear singular perturbation problem

Summary We present a difference scheme for solving a semilinear singular perturbation problem with any number of turning points of arbitrary orders. It is shown that a solution of the scheme converges, uniformly in a perturbation parameter, to that of the continuous problem.     

A Taylor series method for the numerical solution of two-point boundary value problems

Summary For the numerical solution of two-point boundary value problems a shooting algorithm based on a Taylor series method is developed. Series coefficients are generated automatically by recurrence formulas. The performance of the algorithm is demonstrated by solving six problems arising in nonlinear shell theory, chemistry and superconductivity.     

Improvement of numerical solution of boundary layer problems by incorporation of asymptotic approximations

Summary A method for improvement of the numerical solution of differential equations by incorporation of asymptotic approximations is investigated for a class of singular perturbation problems.Uniform error estimates are derived for this method when implemented in known difference schemes and applied to linear second order O.D.E.'s. An improvement by a factor of ⁿ⁺¹ can be obtained (where is the small parameter andn is the order of the asymptotic approximation) for a small amount of extra work. Numerical experiments are presented.     

High order methods for the numerical integration of ordinary differential equations

Summary High order implicit integration formulae with a large region of absolute stability are developed for the approximate numerical integration of both stiff and non-stiff systems of ordinary differential equations. The algorithms derived behave essentially like one step methods and are demonstrated by direct application to certain particular examples.     

High orderP-stable formulae for the numerical integration of periodic initial value problems

Summary Recently there has been considerable interest in the approximate numerical integration of the special initial value problemy=f(x, y) for cases where it is known in advance that the required solution is periodic. The well known class of Störmer-Cowell methods with stepnumber greater than 2 exhibit orbital instability and so are often unsuitable for the integration of such problems. An appropriate stability requirement for the numerical integration of periodic problems is that ofP-stability. However Lambert and Watson have shown that aP-stable linear multistep method cannot have an order of accuracy greater than 2. In the present paper a class of 2-step methods of Runge-Kutta type is discussed for the numerical solution of periodic initial value problems.P-stable formulae with orders up to 6 are derived and these are shown to compare favourably with existing methods.     

Enlarging the domain of convergence for multiple shooting by the homotopy method

Summary The homotopy method is a frequently used technique in overcoming the local convergence nature of multiple shooting. In this paper sufficient conditions are given that guarantee the homotopy process to be feasible. The results are applicable to a class of two-point boundary value problems. Finally, the numerical solution of two practical problems arising in physiology is described.     

Finite difference methods and their convergence for a class of singular two point boundary value problems

Summary We discuss the construction of three-point finite difference approximations and their convergence for the class of singular two-point boundary value problems: (x y)=f(x,y), y(0)=A, y(1)=B, 0<<1. We first establish a certain identity, based on general (non-uniform) mesh, from which various methods can be derived. To obtain a method having order two for all (0,1), we investigate three possibilities. By employing an appropriate non-uniform mesh over [0,1], we obtain a methodM ₁ based on just one evaluation off. For uniform mesh we obtain two methodsM ₂ andM ₃ each based on three evaluations off. For =0,M ₁ andM ₂ both reduce to the classical second-order method based on one evaluation off. These three methods are investigated, theirO(h ²)-convergence established and illustrated by numerical examples.     

Improved difference schemes for the Dirichlet problem of Poisson's equation in the neighbourhood of corners

Summary The standard 5-point difference scheme for the model problem u=f on a special polygonal domain with given boundary values is modified at a few points in the neighbourhood of the corners in such a way that the order of convergence at interior points is the same as in the case of a smooth boundary. As a side result improved error bounds for the usual method in the neighbourhood of corners are given.     

On a mixed finite element approximation of the Stokes problem (I)

Summary We study in this paper a new mixed finite element approximation of the Stokes' problem in the velocity pressure formulation. This approximation which is based on a new variational principle allows the use of low order Lagrange elements and leads to optimal order of convergence for the velocity and the pressure. Iterative and direct methods for the solution of the approximate problems will be discussed in a forthcoming paper.     

On the stability of backward differentiation methods

Summary The approximation of linear systemsy=–A(t)y+b(t) by backward differentiation methods up to order 5 is considered. It is proved that the error does not increase if the real symmetric matrixA(t) is positive definite andA(t) is negative semi-definite.     

On the reduction of Holt's problem to a finite interval

Summary The reduction of Holt's problem to a finite interval is studied. We give the error estimate for the reduction. The solution of the boundary value problem is computed by a variant of the chasing method.This work was partly supported by the Hungarian Academic Research Foundation under project N⁰1-3-86-221     

Order conditions for numerical methods for partitioned ordinary differential equations

Summary Motivated by the consideration of Runge-Kutta formulas for partitioned systems, the theory of P-series is studied. This theory yields the general structure of the order conditions for numerical methods for partitioned systems, and in addition for Nyström methods fory=f(y,y), for Rosenbrock-type methods with inexact Jacobian (W-methods). It is a direct generalization of the theory of Butcher series [7, 8]. In a later publication, the theory ofP-series will be used for the derivation of order conditions for Runge-Kutta-type methods for Volterra integral equations [1].     

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     

Numerical computation of branch points in ordinary differential equations

Summary This paper deals with the computation of branch points in ordinary differential equations. A direct numerical method is presented which requires the solution of only one boundary value problem. The method handles the general case of branching from a nontrivial solution which is a-prioriunknown. A testfunction is proposed which may indicate branching if used in continuation methods. Several real-life problems demonstrate the procedure.     

On improving the absolute stability of local extrapolation

Summary A widely used technique for improving the accuracy of solutions of initial value problems in ordinary differential equations is local extrapolation. It is well known, however, that when using methods appropriate for solving stiff systems of ODES, the stability of the method can be seriously degraded if local extrapolation is employed. This is due to the fact that performing local extrapolation on a low order method is equivalent to using a higher order formula and this high order formula may not be suitable for solving stiff systems. In the present paper a general approach is proposed whereby the correction term added on in the process of local extrapolation is in a sense a rational, rather than a polynomial, function. This approach allows high order formulae with bounded growth functions to be developed. As an example we derive anA-stable rational correction algorithm based on the trapezoidal rule. This new algorithm is found to be efficient when low accuracy is requested (say a relative accuracy of about 1%) and its performance is compared with that of the more familiar Richardson extrapolation method on a large set of stiff test problems.     

On the design of high order exponentially fitted formulae for the numerical integration of stiff systems

Summary Single step exponentially fitted integration formulae of orders 4 and 6 are derived. The final approximation to the required solution is obtained via a linear combination of asymptotically less accurate solutions obtained using conventional implicit integration formulae. This linear combination is performed in such a way that the final integration formula, as well as having an increased order of accuracy, integrates a certain pre-determined ordinary differential equation exactly. The algorithms developed are illustrated by means of some numerical examples.     

Numerische Behandlung von Verzweigungsproblemen bei gewöhnlichen Differentialgleichungen

Summary We present a new method for the numerical solution of bifurcation problems for ordinary differential equations. It is based on a modification of the classical Ljapunov-Schmidt-theory. We transform the problem of determining the nontrivial branch bifurcating from the trivial solution into the problem of solving regular nonlinear boundary value problems, which can be treated numerically by standard methods (multiple shooting, difference methods).

     

On numerical asymptotic behavior of finite element solutions for parabolic equations

Summary In this paper, we investigate the numerical asymptotic behavior of the finite element solutions for linear parabolic equations under some appropriate conditions. We also give some results of numerical experiments in the two dimensional problems to indicate the effectiveness of our results.     

Einschließungsaussagen bei Differentialoperatoren zweiter Ordnung durch punktweise Ungleichungen

Summary We consider non-inverse positive differential operators of the second order and show that error estimates can be obtained through methods of inverse positivity alone. This means a significant simplification if the operator has only one negative eigenvalue. In addition we show how to obtain Range-Domain implications for operators with mixed boundary conditions.

The author would like to thank the European Research Office of the United States Army for their kind interest