Error analysis of the combination technique

Summary. The combination technique is a method to reduce the computational time in the numerical approximation of partial differential equations. In this paper, we present a new technique to analyze the convergence rate of the combination technique. This technique is applied to general second order elliptic differential equations in two dimensions. Furthermore, it is proved that the combination technique for Poisson's equation convergences in arbitrary dimensions. Received September 25, 1997 / Revised version received October 22, 1998 / Published online September 7, 1999     

On the Existence of Star Products on Quotient Spaces of Linear Hamiltonian Torus Actions

We discuss BFV deformation quantization (Bordemann et al. in A homological approach to singular reduction in deformation quantization, singularity theory, pp. 443–461. World Scientific, Hackensack, 2007) in the special case of a linear Hamiltonian torus action. In particular, we show that the Koszul complex on the moment map of an effective linear Hamiltonian torus action is acyclic. We rephrase the nonpositivity condition of Arms and Gotay (Adv Math 79(1):43–103, 1990) for linear Hamiltonian torus actions. It follows that reduced spaces of such actions admit continuous star products.      

Cyclic cocycles on deformation quantizations and higher index theorems

We construct a nontrivial cyclic cocycle on the Weyl algebra of a symplectic vector space. Using this cyclic cocycle we construct an explicit, local, quasi-isomorphism from the complex of differential forms on a symplectic manifold to the complex of cyclic cochains of any formal deformation quantization thereof. We give a new proof of Nest-Tsygan's algebraic higher index theorem by computing the pairing between such cyclic cocycles and the K-theory of the formal deformation quantization. Furthermore, we extend this approach to derive an algebraic higher index theorem on a symplectic orbifold. As an application, we obtain the analytic higher index theorem of Connes-Moscovici and its extension to orbifolds.     

Anomalous roughness evolution of rubrene thin films observed in real time during growth

We study the growth and structure of thin films of the organic semiconductor rubrene during organic molecular beam deposition (OMBD) on silicon oxide in situ and in real time using X-ray scattering. Using in situ grazing incidence diffraction (GID) we find a small degree of local order but an otherwise largely disordered structure, consistent with out of plane scans. Monitoring the surface morphology in real time during growth, we find relatively smooth films (surface roughness sigma below approximately 15 A for thicknesses up to at least 600 A) and a significant delay before the onset of roughening. This anomalous roughening in the beginning and crossover to normal roughening later during growth may be related to conformational changes of rubrene in the early stages of growth.     

An iterative solver for the finite‐difference frequency‐domain (FDFD) method for the simulation of materials with negative permittivity

A stable iterative solver for the simulation of optical waves in metals using finite difference frequency domain (FDFD) method is presented. The corresponding discretization of Maxwell's equations enables simulating electromagnetic waves in structures when materials with negative permittivity are involved. Convergence of the iterative solver is proved for positive and negative permittivities. Numerical results are presented for a thin‐film silicon solar cell structure containing silver back contact. Copyright © 2010 John Wiley & Sons, Ltd.     

A multilevel algorithm for the solution of second order elliptic differential equations on sparse grids

A multilevel algorithm is presented that solves general second order elliptic partial differential equations on adaptive sparse grids. The multilevel algorithm consists of several V-cycles in - and -direction. A suitable discretization provide that the discrete equation system can be solved in an efficient way. Numerical experiments show a convergence rate of order for the multilevel algorithm. Received April 19, 1996 / Revised version received December 9, 1996     

On Continuous Hochschild comology and Cohomology Groups

Using the method of continuous projective resolutions established by Alain Connes, we calculate the continuous Hochschild homology and cohomology groups of the Fréchet algebra of smooth functions on a manifold M. These (co)homology groups are allowed to take values in the Fréchet space E of smooth sections of a vector bundle E over M and in the strong dual E*. Moreover, we show that in the cohomological case the differential and continuous Hochschild cohomologies are naturally isomorphic.     

Two-dimensional dynamic simulations of DFB lasers and MOPAs

Due to the complex lateral structure of Distributed Feedback lasers (DFB) and Master Oscillator Power Amplifiers (MOPAs), one-dimensional methods like the classical Transfer Matrix Method (TMM) are not suitable for simulating the optical wave in these devices. Therefore, we applied Trigonometric Finite Wave Elements (TFWE) that generalize the TMM in two or three dimensions. By coupling the dynamic wave equation with simplified temperature and drift-diffusion models, we can simulate the dynamic behavior of DFB lasers and MOPAs. Furthermore, by Fourier transformation, we can calculate the modes of the laser cavity and the corresponding wavelengths. By this approach, the influence of injection current and temperature on the resulting modes and output power can be analyzed in detail.     

Absorption line strengths of <Superscript>15</Superscript>NH<Subscript>3</Subscript> in the near infrared spectral region

We present initial results of an investigation of the near infrared absorption spectrum of ¹⁵NH₃ between 6468 and 6692 cm⁻¹. A widely tunable external cavity diode laser is used in a direct absorption setup to determine the line positions and line strengths of several lines in that spectral range. Line data measurements on a ¹⁴NH₃ sample are used for validation of the setup by comparison of the results with available literature data. The presented overview measurements on absorption lines of ¹⁵NH₃ have been performed to serve as a starting point for candidate line selection for prospective isotopic ratio measurements of ¹⁴NH₃ and ¹⁵NH₃.