A note on initial higher‐order convergence results for boundary element methods with approximated boundary conditions

We consider a variational problem associated with a pseudo‐differential operator of negative order 2s < 0 with an additional approximation of the given linear form. Such an approximation may correspond to an interpolation of given boundary conditions for a partial differential equation. The asymptotic order of convergence of the related Galerkin solution can be reached for ν = μ +2s, where ν and μ are the polynomial degrees of the trial functions used to approximate the solution and boundary conditions, respectively. The main result of this article is to prove that one can expect higher initial rates in the convergence behavior, even in the worst case of isoparametric approximations (ν = μ) when the error is measured in the Sobolev norm H^τ(Γ) with τ ∈ [s, 0]; i.e., this initial estimate is also valid in the energy norm ‖ · ‖. This result is based on the relation between the approximation error of the Galerkin solution without this additional approximation and the additional approximation error itself. As an illustration of the technique, an application of a boundary element method for the Dirichlet problem of a second‐order elliptic partial differential operator is given. Numerical examples confirm the theoretical results for this case. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 581–588, 2000.     

Synthesis and Molecular Structure of a Tripodal Triaminostibine

Reaction of the tripodal trilithium triamide [MeSi{SiMe₂N(Li)(p‐Tol)}₃(thf)₂] ( 1 ) with SbCl₃ in toluene gave the corresponding triaminostibine [MeSi{SiMe₂N(p‐Tol)}₃Sb] ( 2 ) in good yield. Its [2,2,2]bicyclooctane‐related cage structure, comprising the trisilylsilane unit and the triamidostibine fragment, was established by a single crystal X‐ray structure analysis: Space group P1, Z = 2, lattice dimensions at 293 K: a = 8.645(4), b = 10.029(5), c = 19.678(9) Å, α = 98.50(3)°, β = 97.46(3)°, γ = 94.80(3)°, R = 0.0216.     

Synthese von Sulfolenobilinen und deren Cyclisierung zu Chlorinatozink‐Fulleren‐Dyaden

Synthesis of Sulfolenobilins and Their Cyclization Directed to Chlorinatozinc‐Fullerene Dyads A novel chlorinatozinc‐fullerene dyad 18 was synthesized to model the photosynthetic reaction center. The synthetic key step for the formation of the dyad 18 is an unusual one‐pot reaction of the (sulfolenobilinato)nickel rac‐ 16a , b with concomitant generation of the chlorin macrocycle and linkage to the [5,6]fullerene‐C₆₀‐I_h. This one‐pot reaction is a complex cascade of single reaction steps with a total yield of 32% and an average yield of 83% for the individual steps. The chlorinatozinc‐fullerene dyad is so far one of three examples that contain chlorin moieties, the chromophores in naturally occurring photosynthetic systems.     

New side-chain poly(methacryl-rotaxanes) bearing cyclodextrins as non-covalently anchored ring components. Chemoenzymatic synthesis and degradation

The chemoenzymatic synthesis of new side-chain polyrotaxanes ( 6 ) bearing non-covalently attached cyclodextrins is described. Starting from 11-methacryloylaminoundecanoic acid, up to 8 units of 12-hydroxylauric acid were condensed in the presence of a lipase from Candida antarctica. They were copolymerized with methyl methacrylate to comb-like polymers ( 4 ) bearing free carboxylic groups at the end of the oligoester side chains. These free carboxylic groups were then condensed chemically with a nucleophilic barrier group, N-(4-aminobutanoyl)-4-triphenylmethylaniline ( 5 ), in the presence of dimethylated β-cyclodextrin, yielding new polyrotaxanes that contain statistically more than 1 ring component per side chain.     

ADOL‐C: Automatic Differentiation Using Operator Overloading in C++

In this paper, we present two strategies for the implementation of Automatic Differentiation (AD) based on the operator overloading facility in C++. Subsequently, we describe the capabilities of the AD‐tool ADOL‐C that applies operator overloading to differentiate C‐ and C++‐code. Finally, we discuss some applications of ADOL‐C.     

N,N-Dimethylglycinatokomplexe von Platin(IV)

N,N -Dimethylglycinato Complexes of Platinum(IV) The aquapentachloroplatinic acid (H₃O)-[PtCl₅(H₂O)] · 2(18-cr-6) · 6 H₂O ( 1 ) reacts with N,N-dimethylglycine (Me₂glyH) to give cis-[PtCl₂(N,O-Me₂gly)₂] · (18-cr-6) ( 6 ) and (Me₂glyH₂)[PtCl₄(N,O-Me₂gly)] ( 7 ). Complexes 6 and 7 are characterized by microanalysis, ¹H-NMR and IR spectroscopy as well as by X-ray structure analysis. In both complexes the N,N-dimethylglycinato ligands are N,O-coordinated. In 6 , the amino groups are mutually trans and the carboxylato groups are cis (configuration index: OC-6–22). In the crystal, there are only weak C–H…O interactions between the N-methyl groups of the [PtCl₂(N,O-Me₂gly)₂] complex and the oxygen atoms of the crown ether (shortest C…O contacts: 3.10(2) Å and 3.21(2) Å). In the solid state, 7 exhibits strong cation-anion interactions: The carboxyl group of the cation (Me₂glyH₂)⁺ forms a strong O–H…O bridge to the exocyclic oxygen atom of the carboxylate group of the glycinato ligand (O…O 2.61(1) Å).