Sparse-grid finite-volume multigrid for 3D-problems

We introduce a multigrid algorithm for the solution of a second order elliptic equation in three dimensions. For the approximation of the solution we use a partially ordered hierarchy of finite-volume discretisations. We show that there is a relation with semicoarsening and approximation by more-dimensional Haar wavelets. By taking a proper subset of all possible meshes in the hierarchy, a sparse grid finite-volume discretisation can be constructed.The multigrid algorithm consists of a simple damped point-Jacobi relaxation as the smoothing procedure, while the coarse grid correction is made by interpolation from several coarser grid levels.The combination of sparse grids and multigrid with semi-coarsening leads to a relatively small number of degrees of freedom,N, to obtain an accurate approximation, together with anO(N) method for the solution. The algorithm is symmetric with respect to the three coordinate directions and it is fit for combination with adaptive techniques.To analyse the convergence of the multigrid algorithm we develop the necessary Fourier analysis tools. All techniques, designed for 3D-problems, can also be applied for the 2D case, and — for simplicity — we apply the tools to study the convergence behaviour for the anisotropic Poisson equation for this 2D case.     

Approximate sine-Gordon solitons

We look at the recently proposed scheme of approximating a sine-Gordon soliton by an expression derived from two dimensional instantons. We point out that the scheme of Sutcliffe in which he uses two dimensional instantons can be generalised to higher dimensions and that these generalisations produce even better approximations than the original approximation. We also comment on generalisations to other models.     

Algorithms for ham-sandwich cuts

Given disjoint setsP ₁,P ₂, ...,P _d inR ^d withn points in total, ahamsandwich cut is a hyperplane that simultaneously bisects theP _i . We present algorithms for finding ham-sandwich cuts in every dimensiond>1. Whend=2, the algorithm is optimal, having complexityO(n). For dimensiond>2, the bound on the running time is proportional to the worst-case time needed for constructing a level in an arrangement ofn hyperplanes in dimensiond−1. This, in turn, is related to the number ofk-sets inR ^d−1 . With the current estimates, we get complexity close toO(n ^3/2 ) ford=3, roughlyO(n ^8/3 ) ford=4, andO(n ^d−1−a(d) ) for somea(d)>0 (going to zero asd increases) for largerd. We also give a linear-time algorithm for ham-sandwich cuts inR ³ when the three sets are suitably separated. A preliminary version of the results of this paper appeared in [16] and [17]. Part of this research by J. Matoušek was done while he was visiting the School of Mathematics, Georgia Institute of Technology, Atlanta, and part of his work on this paper was supported by a Humboldt Research Fellowship. W. Steiger expresses gratitude to the NSF DIMACS Center at Rutgers, and his research was supported in part by NSF Grants CCR-8902522 and CCR-9111491.     

Novel optical techniques for the analysis of polymer surfaces and thin films

Summary We report on novel optical techniques, based on evanescent waves, for the characterization of polymer surfaces and thin films. We first describe photo-ablation studies with polysilane films investigated by surface plasmon microscopy, a technique which is particularly well-suited for ultrathin samples. Thicker films that are homogeneous enough to carry optical waveguide modes can be characterized with high lateral resolution by the recently developed waveguide microscopy. We demonstrate this for a thin film of a solid polyelectrolyte. Finally, we report on surface plasmon field-enhanced Raman-spectroscopic and -imaging investigations of ultrathin Langmuir-Blodgett-Kuhn-layers of cadmium arachidate.W. Hickel is now with HOECHST AG, Angewandte Physik, W-6230 Frankfurt 80     

Electron-stimulated desorption of D (H) from condensed D2O (H2O) films

The electron-stimulated desorption (ESD) of D⁻ and H⁻ ions from condensed D₂O and H₂O films is investigated. Three low-energy peaks are observed in the ESD anion yield, which are identified as arising from excitation of ²B₁, ²A₁ and ²B₂ dissociative electron attachment (DEA) resonances. Additional structure is observed between 18 and 32 eV, which may be due to ion pair formation or to DEA resonances involving the 2a₁ orbital. The ion yield resulting from excitation of the ²B₁ resonance increases as the film is heated. We attribute the increase in the ion yield to thermally induced hydrogen bond breaking near the surface, which enhances the lifetimes of the excited states that lead to desorption.     

Existence of solutions for the time-dependent thermistor equations

The equations describing the behaviour of temperature/flow microsensorsemploy-ing polysilicon thermistors form a degenerate nonlinearparabolic system. One special feature is that different equationsof this system are to be satisfied on different domains. Bymeans of supersolutions and subsolutions, the maximum principle,and fixed point arguments, the authors show the existence ofsolutions to the equations.     

On the retrograde condensation of one or two liquids

This contribution discusses the phenomena of retrograde condensation of one or two liquids. It w1 be shown that both phenomena can be well understood. Also the relation of retrograde condensation of one liquid phase with the condensation behavior of natural gas will be discussed. Similarly that of two liquid phases with multiple phase behavior occurring in low temperature reservoir fluids will be pointed out.