Uniformly convergent difference schemes for singularly perturbed parabolic diffusion-convection problems without turning points

We consider first the initial-boundary value problem for the parabolic equation

A method for computing the tension parameters in convexity-preserving spline-in-tension interpolation

Summary This paper is concerned with the problem of convexity-preservng (orc-preserving) interpolation by using Exponential Splines in Tension (or EST's). For this purpose the notion of ac-preserving interpolant, which is usually employed in spline-in-tension interpolation, is refined and the existence ofc-preserving EST's is established for the so-calledc-admissible data sets. The problem of constructing ac-preserving and visually pleasing EST is then treated by combining a generalized Newton-Raphson method, due to Ben-Israel, with a step-length technique which serves the need for visual pleasantness. The numerical performance of the so formed iterative scheme is discussed for several examples.     

Uniform fourth order difference scheme for a singular perturbation problem

Summary The numerical solution of a nonlinear singularly perturbed two-point boundary value problem is studied. The developed method is based on Hermitian approximation of the second derivative on special discretization mesh. Numerical examples which demonstrate the effectiveness of the method are presented.This research was partly supported by NSF and SIZ for Science of SAP Vojvodina through funds made available to the U.S.-Yugoslav Joint Board on Scientific and Technological Cooperation (grants JF 544, JF 799)     

Continued fractions associated with the Newton-Padé table

Summary We discuss first the block structure of the Newton-Padé table (or, rational interpolation table) corresponding to the double sequence of rational interpolants for the data{(z _k, h(z_k)} _k =0. (The (m, n)-entry of this table is the rational function of type (m,n) solving the linearized rational interpolation problem on the firstm+n+1 data.) We then construct continued fractions that are associated with either a diagonal or two adjacent diagonals of this Newton-Padé table in such a way that the convergents of the continued fractions are equal to the distinct entries on this diagonal or this pair of diagonals, respectively. The resulting continued fractions are generalizations of Thiele fractions and of Magnus'sP-fractions. A discussion of an some new results on related algorithms of Werner and Graves-Morris and Hopkins are also given.Dedicated to the memory of Helmut Werner (1931–1985)     

A fourth order method for a singular two-point boundary value problem

Recently, Chawla et al. described a second order finite difference method for the class of singular two-point boundary value problems:

Stability properties of collocation methods

Lower bounds for are given for which equidistant s-point collocation methods areA()-stable for arbitrarys.     

Numerical methods for reaction-diffusion problems with non-differentiable kinetics

Summary We consider a class of steady-state semilinear reaction-diffusion problems with non-differentiable kinetics. The analytical properties of these problems have received considerable attention in the literature. We take a first step in analyzing their numerical approximation. We present a finite element method and establish error bounds which are optimal for some of the problems. In addition, we also discuss a finite difference approach. Numerical experiments for one- and two-dimensional problems are reported.Dedicated to Ivo Babuka on his sixtieth birthdayResearch partially supported by the Air Force Office of Scientific Research, Air Force Systems Command, USAF under Grant Number AFOSR 85-0322     

On theR-order of coupled sequences arising in single-step type methods

Summary As shown in preceding papers of the authors, the verification of anR-convergence order for sequences coupled by a system (1.1) of basic inequalities can be reduced to the positive solvability of system (3.3) of linear inequalities. Further, the bestR-order implied by (1.1) is equal to the minimal spectral radius of certain matrices composed from the exponents occuring in (1.1). Now, these results are proven in a unified and essentially simpler way. Moreover, they are somewhat extended in order to facilitate applications to concrete methods.     

Adaptive boundary element methods for strongly elliptic integral equations

Summary Integral operators are nonlocal operators. The operators defined in boundary integral equations to elliptic boundary value problems, however, are pseudo-differential operators on the boundary and, therefore, provide additional pseudolocal properties. These allow the successful application of adaptive procedures to some boundary element methods. In this paper we analyze these methods for general strongly elliptic integral equations and obtain a-posteriori error estimates for boundary element solutions. We also apply these methods to nodal collocation with odd degree splines. Some numerical examples show that these adaptive procedures are reliable and effective.This work was carried out while Dr. De-hao Yu was an Alexander-von-Humboldt-Stiftung research fellow at the University of Stuttgart in 1987, 1988     

Convergence of mixed finite element approximations for the shallow arch problem

Summary The stability and convergence of mixed finite element methods are investigated, for an equilibrium problem for thin shallow elastic arches. The problem in its standard form contains two terms, corresponding to the contributions from the shear and axial strains, with a small parameter. Lagrange multipliers are introduced, to formulate the problem in an alternative mixed form. Questions of existence and uniqueness of solutions to the standard and mixed problems are addressed. It is shown that finite element approximations of the mixed problem are stable and convergent. Reduced integration formulations are equivalent to a mixed formulation which in general is distinct from the formulation shown to be stable and convergent, except when the order of polynomial interpolationt of the arch shape satisfies 1tmin (2,r) wherer is the order of polynomial approximation of the unknown variables.