The optimal convergence of the h–p version of the boundary element method with quasiuniform meshes for elliptic problems on polygonal domains

In the framework of the Jacobi-weighted Besov spaces, we analyze the lower and upper bounds of errors in the h–p version of boundary element solutions on quasiuniform meshes for elliptic problems on polygons. Both lower bound and upper bound are optimal in h and p, and they are of the same order. The optimal convergence of the h–p version of boundary element method with quasiuniform meshes is proved, which includes the optimal rates for h version with quasiuniform meshes and the p version with quasiuniform degrees as two special cases. Dedicated to Professor Charles Micchelli on the occasion of his sixtieth birthday Mathematics subject classification (2000) 65N38. Benqi Guo: The work of this author was supported by NSERC of Canada under Grant OGP0046726 and was complete during visiting Newton Institute for Mathematical Sciences, Cambridge University for participating in special program “Computational Challenges in PDEs” in 2003. Norbert Heuer: This author is supported by Fondecyt project No. 1010220 and by the FONDAP Program (Chile) on Numerical Analysis. Current address: Mathematical Sciences, Brunel University, Uxbridge, U.K.     

Angular vibrations of cryogenically cooled double‐crystal monochromators

The effect of angular vibrations of the crystals in cryogenically cooled monochromators on the beam performance has been studied theoretically and experimentally. A simple relation between amplitude of the vibrations and size of the focused beam is developed. It is shown that the double‐crystal monochromator vibrations affect not only the image size but also the image position along the optical axis. Several methods to measure vibrations with the X‐ray beam are explained and analyzed. The methods have been applied to systematically study angular crystal vibrations at monochromators installed at the PETRA III light source. Characteristic values of the amplitudes of angular vibrations for different monochromators are presented.     

Nonsingularity of some classes of matrices, and minimal solutions of Silverman''s game on discrete sets

We establish the irreducibility of each game in four infinite three-parameter families of even order Silverman games, and the major step in doing so is to prove that certain matrices A, related in a simple way to the payoff matrices, are nonsingular for all relevant values of the parameters. This nonsingularity is established by, in effect, producing a matrix D such that AD is known to be nonsingular. The elements of D are polynomials from six interrelated sequences of polynomials closely related to the Chebyshev polynomials of the second kind. Each of these sequences satisfies a second order recursion, and consequently has many Fibonacci-like properties, which play an essential role in proving that the product AD is what we claim it is. The matrices D were found experimentally, by discovering patterns in low order cases worked out with the help of some computer algebra systems. The corresponding results for four families of odd order games were reported in an earlier paper.     

On the numerical analysis of nonlinear twofold saddle point problems

We provide a general abstract theory for the solvability andGalerkin approximation of nonlinear twofold saddle point problems.In particular, a Strang error estimate containing the consistencyterms arising from the approximation of the continuous operatorsinvolved is deduced. Then we apply these results to analysea fully discrete Galerkin scheme for a twofold saddle pointformulation of a nonlinear elliptic boundary value problem indivergence form. Some numerical results are also presented.     

New dicarboxylic acid bipyridine ligand for ruthenium polypyridyl sensitization of TiO2

An ambidentate dicarboxylic acid bipyridine ligand, (4,5-diazafluoren-9-ylidene) malonic acid (dfm), was synthesized for coordination to Ru(II) and mesoporous nanocrystalline (anatase) TiO(2) thin films. The dfm ligand provides a conjugated pathway from the pyridyl rings to the carbonyl carbons of the carboxylic acid groups. X-ray crystal structures of [Ru(bpy)(2)(dfm)]Cl(2) and the corresponding diethyl ester compound, [Ru(bpy)(2)(defm)](PF(6))(2), were obtained. The compounds displayed intense metal-to-ligand charge transfer (MLCT) absorption bands in the visible region (ε > 11,000 M(-1) cm(-1) for [Ru(bpy)(2)(dfm)](PF(6))(2) in acetonitrile). Significant room temperature photoluminescence, PL, was absent in CH(3)CN but was observed at 77 K in a 4:1 EtOH:MeOH (v:v) glass. Cyclic voltammetry measurements revealed quasi-reversible Ru(III/II) electrochemistry. Ligand reductions were quasi-reversible for the diethyl ester compound [Ru(bpy)(2)(defm)](2+), but were irreversible for [Ru(bpy)(2)(dfm)](2+). Both compounds were anchored to TiO(2) thin films by overnight reactions in CH(3)CN to yield saturation surface coverages of 3 × 10(-8) mol/cm(2). Attenuated total reflection infrared measurements revealed that the [Ru(bpy)(2)(dfm)](2+) compound was present in the deprotonated carboxylate form when anchored to the TiO(2) surface. The MLCT excited states of both compounds injected electrons into TiO(2) with quantum yields of 0.70 in 0.1 M LiClO(4) CH(3)CN. Micro- to milli-second charge recombination yielded ground state products. In regenerative solar cells with 0.5 M LiI/0.05 M I(2) in CH(3)CN, the Ru(bpy)(2)(dfm)/TiO(2) displayed incident photon-to-current efficiencies of 0.7 at the absorption maximum. Under the same conditions, the diethylester compound was found to rapidly desorb from the TiO(2) surface.     

Coupling of mixed finite element and stabilized boundary element methods for a fluid–solid interaction problem in 3D

We introduce and analyze the coupling of a mixed finite element and a boundary element for a three‐dimensional time‐harmonic fluid–solid interaction problem. We consider a formulation in which the Cauchy stress tensor and the rotation are the main variables in the elastic structure and use the usual pressure formulation in the acoustic fluid. The mixed variational formulation in the solid is completed with boundary integral equations relating the Cauchy data of the acoustic problem on the coupling interface. A crucial point in our formulation is the stabilization technique introduced by Hiptmair and coworkers to avoid the well‐known instability issue appearing in the boundary element method treatment of the exterior Helmholtz problem. The main novelty of this formulation, with respect to a previous approach, consists in reducing the computational domain to the solid media and providing a more accurate treatment of the far field effect. We show that the continuous problem is well‐posed and propose a conforming Galerkin method based on the lowest‐order Arnold–Falk–Winther mixed finite element. Finally, we prove that the numerical scheme is convergent with optimal order.Copyright © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1211–1233, 2014     

A posteriori error analysis for a boundary element method with nonconforming domain decomposition

We present and analyze an a posteriori error estimator based on mesh refinement for the solution of the hypersingular boundary integral equation governing the Laplacian in three dimensions. The discretization under consideration is a nonconforming domain decomposition method based on the Nitsche technique. Assuming a saturation property, we establish quasireliability and efficiency of the error estimator in comparison with the error in a natural (nonconforming) norm. Numerical experiments with uniform and adaptively refined meshes confirm our theoretical results. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 947–963, 2014     

The structure and ordering of ε-MnO2

The presence of ε-MnO₂ as a major component of electrolytic manganese dioxide (EMD) has been demonstrated by a combined X-ray diffraction/transmission electron microscopy (TEM) study. ε-MnO₂ usually has a partially ordered defect NiAs structure containing 50% cation vacancies; these vacancies can be fully ordered by a low temperature (200 °C) heat treatment to form a pseudohexagonal but monoclinic superlattice.Numerous fine-scale anti-phase domain boundaries are present in ordered ε-MnO₂ and cause extensive peak broadening and a massive shift of a very intense, 0.37 nm superlattice peak. This suggests a radically different explanation of the ubiquitous, very broad ∼0.42 nm peak (∼21-22° 2θ, CuKα radiation) in EMDs, which heretofore has been attributed to Ramsdellite containing numerous planar defects. This work confirms the multi-phase model of equiaxed EMDs proposed by Heuer et al. [ITE Lett. 1(6) (2000) B50; Proc. Seventh Int. Symp. Adv. Phys. Fields 92 (2001)], rather than the defective single-phase model of Chabre and Pannetier [Prog. Solid State Chem. 23 (1995) 1] and Bowden et al. [ITE Lett. 4(1) (2003) B1].