Winfried Auzinger

An Elimination Algorithm for the Computation of All Zeros of a System of Multivariate Polynomial Equations

A direct numerical method is proposed for the determination of all isolated zeros of a system of multivariate polynomial equations. By “polynomial combination”, the system is reduced to a special form which may be interpreted as a multiplication table for power products modulo the system. The zeros are then formed from an ordinary eigenvalue problem for the matrix of the multiplication table. Degenerate situations may be handled by perturbing them into general form and reaching the zeros of the unperturbed system via a homotopy method.

Efficient Collocation Schemes for Singular Boundary Value Problems

We discuss an error estimation procedure for the global error of collocation schemes applied to solve singular boundary value problems with a singularity of the first kind. This a posteriori estimate of the global error was proposed by Stetter in 1978 and is based on the idea of Defect Correction, originally due to Zadunaisky. Here, we present a new, carefully designed modification of this error estimate which not only results in less computational work but also appears to perform satisfactorily for singular problems. We give a full analytical justification for the asymptotical correctness of the error estimate when it is applied to a general nonlinear regular problem. For the singular case, we are presently only able to provide computational evidence for the full convergence order, the related analysis is still work in progress. This global estimate is the basis for a grid selection routine in which the grid is modified with the aim to equidistribute the global error. This procedure yields meshes suitable for an efficient numerical solution. Most importantly, we observe that the grid is refined in a way reflecting only the behavior of the solution and remains unaffected by the unsmooth direction field close to the singular point.

Analysis of a New Error Estimate for Collocation Methods Applied to Singular Boundary Value Problems

We discuss an a posteriori error estimate for the numerical solution of boundary value problems for nonlinear systems of ordinary differential equations with a singularity of the first kind. The estimate for the global error of an approximation obtained by collocation with piecewise polynomial functions is based on the defect correction principle. We prove that for collocation methods which are not superconvergent, the error estimate is asymptotically correct. As an essential prerequisite we derive convergence results for collocation methods applied to nonlinear singular problems.

A Collocation Code for Singular Boundary Value Problems in Ordinary Differential Equations

We present a MATLAB package for boundary value problems in ordinary differential equations. Our aim is the efficient numerical solution of systems of ODEs with a singularity of the first kind, but the solver can also be used for regular problems. The basic solution is computed using collocation methods and a new, efficient estimate of the global error is used for adaptive mesh selection. Here, we analyze some of the numerical aspects relevant for the implementation, describe measures to increase the efficiency of the code and compare its performance with the performance of established standard codes for boundary value problems.

Asymptotic error expansions for stiff equations: an analysis for the implicit midpoint and trapezoidal rules in the strongly stiff case

SummaryThe structure of the global discretization error is studied for the implicit midpoint and trapezoidal rules applied to nonlinearstiff initial value problems. The point is that, in general, the global error contains nonsmooth (oscillating) terms at the dominanth2-level. However, it is shown in the present paper that for special classes of stiff problems these nonsmooth terms contain an additional factor ɛ (where-1/ɛ is the magnitude of the stiff eigenvalues). In these cases a “full” asymptotic error expansion exists in thestrongly stiff case (ε sufficiently small compared to the stepsizeh). The general case (where the oscillating error components areO(h2) and notO(ɛh2)) and applications of our results (extrapolation and defect correction algorithims) will be studied in separate papers.

Asymptotic error expansions for stiff equations: the implicit Euler scheme

This paper discusses the existence of an asymptotic expansion for the global error of the implicit Euler scheme applied to stiff nonlinear systems of ordinary differential equations. It is shown that in strongly stiff situations, a full asymptotic expansion exists at all gridpoints. For the mildly stiff case it is shown that the full order of the remainder term, which inevitably breaks down at the first gridpoints after a stepsize change, reappears at the subsequent gridpoints. Our analysis is based on singular perturbation techniques.

Modified Defect Correction Algorithms for ODEs. Part I: General Theory

The well-known method of Iterated Defect Correction (IDeC) is based on the following idea: Compute a simple, basic approximation and form its defect w.r.t. the given ODE via a piecewise interpolant. This defect is used to define an auxiliary, neighboring problem whose exact solution is known. Solving the neighboring problem with the basic discretization scheme yields a global error estimate. This can be used to construct an improved approximation, and the procedure can be iterated. The fixed point of such an iterative process corresponds to a certain collocation solution. We present a variety of modifications to this algorithm. Some of these have been proposed only recently, and together they form a family of iterative techniques, each with its particular advantages. These modifications are based on techniques like defect quadrature (IQDeC), defect interpolation (IPDeC), and combinations thereof. We investigate the convergence on locally equidistant and nonequidistant grids and show how superconvergent approximations can be obtained. Numerical examples illustrate our considerations. The application to stiff initial value problems will be discussed in Part II of this paper.

Fast stray field computation on tensor grids

A direct integration algorithm is described to compute the magnetostatic field and energy for given magnetization distributions on not necessarily uniform tensor grids. We use an analytically-based tensor approximation approach for function-related tensors, which reduces calculations to multilinear algebra operations. The algorithm scales with N4/3 for N computational cells used and with N2/3 (sublinear) when magnetization is given in canonical tensor format. In the final section we confirm our theoretical results concerning computing times and accuracy by means of numerical examples.

A note on convergence concepts for stiff problems

Most convergence concepts for discretizations of nonlinear stiff initial value problems are based on one-sided Lipschitz continuity. Therefore only those stiff problems that admit moderately sized one-sided Lipschitz constants are covered in a satisfactory way by the respective theory. In the present note we show that the assumption of moderately sized one-sided Lipschitz constants is violated for many stiff problems. We recall some convergence results that are not based on one-sided Lipschitz constants; the concept of singular perturbations is one of the key issues. Numerical experience with stiff problems that are not covered by available convergence results is reported.ZusammenfassungDie meisten Konvergenzkonzepte für Diskretisierungen nichtlinearer steifer Anfangswertprobleme basieren auf dem Begriff der einseitigen Lipschitz-Stetigkeit. Folglich sind durch diese theoretischen Konzepte nur steife Probleme mit moderater einseitiger Lipschitzkonstante abgedeckt. In der vorliegenden Arbeit zeigen wir, daß die Annahme moderater einseitiger Lipschitzkonstanten für viele steife Probleme verletzt ist. Wir weisen auf einige Konvergenzresultate hin, die nicht auf einseitigen Lipschitzkonstanten basieren; die Konzepte der singulären Störungstheorie sind hier von wesentlicher Relevanz. Wir berichten über einige numerische Erfahrungen mit steifen Problemen, die durch keine existierende Konvergenztheorie abgedeckt sind.

New a posteriori error estimates for singular boundary value problems

Abstract In this paper, we discuss the asymptotic properties and efficiency of several a posteriori estimates for the global error of collocation methods. Proofs of the asymptotic correctness are given for regular problems and for problems with a singularity of the first kind. We were also strongly interested in finding out which of our error estimates can be applied for the efficient solution of boundary value problems in ordinary differential equations with an essential singularity. Particularly, we compare estimates based on the defect correction principle with a strategy based on mesh halving.