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.     

On the solution of singular linear systems of algebraic equations by semiiterative methods

Summary For a square matrixT ^n,n , where (I–T) is possibly singular, we investigate the solution of the linear fixed point problemx=T x+c by applying semiiterative methods (SIM's) to the basic iterationx ₀ ⁿ ,x _k T c _k–1+c(k1). Such problems arise if one splits the coefficient matrix of a linear systemA x=b of algebraic equations according toA=M–N (M nonsingular) which leads tox=M ^–1 N x+M ^–1 bT x+c. Even ifx=T x+c is consistent there are cases where the basic iteration fails to converge, namely ifT possesses eigenvalues 1 with ||1, or if =1 is an eigenvalue ofT with nonlinear elementary divisors. In these cases — and also ifx=T x+c is incompatible — we derive necessary and sufficient conditions implying that a SIM tends to a vector which can be described in terms of the Drazin inverse of (I–T). We further give conditions under which is a solution or a least squares solution of (I–T)x=c.Research supported in part by the Alexander von Humboldt-Stiftung     

The many limits of mixed means

The sequences introduced by Carlson (1971) are variants of the Gauss arithmetic geometric sequences (which have been elegantly discussed by D. A. Cox (1984, 1985)). Given (complex)a ₀,b ₀ we define

What do multistep methods approximate?

Summary Standard analysis of multistep methods for ODE's assumes the application of an initialization routine that generates the starting points. Here ak-step method is considered directly as a mappingR ^kn R ⁿ . It is shown to approximate a mapping which is expressible directly in terms of the flow of the vector field. Some useful properties of that mapping are shown and for strictly stable methods these are applied to the question of invariant circles near a hyperbolic periodic solution.     

Convolution quadrature and discretized operational calculus. I

Numerical methods are derived for problems in integral equations (Volterra, Wiener-Hopf equations) and numerical integration (singular integrands, multiple time-scale convolution). The basic tool of this theory is the numerical approximation of convolution integrals

Strukturelle Charakterisierung der Dicyanamid-Komplexe von Kupfer(II), Nickel(II) bzw. Kobalt(II) mit Aminopyridinen

Summary New dicyanamide complexes of the typeM[N(CN)₂]₂ L ₂ (M=Cu or Ni;L=2-, 3-, 4-aminopyridine, 2-amino-5-nitropyridine) and Co[N(CN)₂]₂ (2-amino-5-nitropyridine)₂ were prepared and studied by spectroscopic methods as well as by room-temperature magnetic moments. The results show that the Cu(II) complexes have elongated pseudooctahedral structures while the Ni(II) and Co(II) complexes are octahedral. In most cases the N(CN)₂ groups are in bridging function and through their cyanide nitrogens, or-more rarely-amide and cyanide nitrogens connect the basic structure units into polymeric conglomerates. In the Cu(II) systems exchange coupling is seen from the _eff value or ESR spectrum.

Professor Viktor Gutmann zum 70. Geburtstag gewidmet     

Potentiometric studies on the complexation equilibria between some trivalent lanthanide metal ions and biologically active 2-hydroxy-1-naphthaldehyde thiosemicarbazone (HNATS)

Summary The chelation behaviour of some trivalent lanthanide and yttrium metal ion with biologically active 2-hydroxy-1-naphthaldehyde thiosemicarbazone (HNATS) has been investigated by potentiomotric measurements at 20±0.5°C in 75% (v/v) dioxane-water medium at various ionic strengths of sodium perchlorate. The method of Bjerrum and Calvin, as modified by Irving and Rossotti has been used to find out the values of (average number of ligand bound per metal ion) andpL (free ligand exponent). The formation constants of metal chelates have been computed on a PC-XT computer, using a program patterned after that of Sullivan et al. to give _n values using weighted least squares method. TheS _min values (S _min =²) have been calculated. The order of formation constants of chelates was found to be: La³⁺3+3+3+3+3+3+3+3+3+. The formation constants of the chelates formed have been correlated to size and ionization potentials of the metal ions.

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Potentiometrische Untersuchungen der Komplexierungsgleichgewichte zwischen einigen trivalenten Metallionen und biologisch aktivem 2-Hydroxy-1-naphthaldehyd-thiosemicarbazon (HNATS)

Zusammenfassung Es wurde das Chelierungsverhalten einiger trivalenter Lanthanidenionen mit biologisch aktivem 2-Hydroxy-1-naphthaldehyd-thiosemicarbazon (HNATS) mittels potentiometrischer Messungen bei 20±0.5°C in 75% (v/v) Dioxan-Wasser bei verschiedener Ionenstärke an Natriumperchlorat untersucht. Die Methode nach Bjerrum und Calvin in der Modifikation nach Irving und Rossotti wurde zur Ermittlung der Werte (mittlere Anzahl an Liganden pro Metallion) undpL (Exponent an freiem Liganden) verwendet. Die Komplexbildungskonstanten wurden in Anlehnung an Sullivan et al. an einem PC-XT Computer errechnet. DieS _min-Werte (S _min=²) wurden ebenfalls bestimmt. Die Reihung der Chelatbildungskonstanten war: La³⁺3+3+3+3+3+3+3+3+3+. Die Bildungskonstanten sind mit der Größe und dem Ionisierungspotential der Metallionen zu korrelieren.