A new a posteriori residual error estimator is defined and rigorously analysed for anisotropic tetrahedral finite element meshes. All considerations carry over to anisotropic triangular meshes with minor changes only. The lower error bound is obtained by means of bubble functions and the corresponding anisotropic inverse inequalities. In order to prove the upper error bound, it is vital that an anisotropic mesh corresponds to the anisotropic function under consideration. To measure this correspondence, a so-called matching function is defined, and its discussion shows it to be a useful tool. With its help anisotropic interpolation estimates and subsequently the upper error bound are proven. Additionally it is pointed out how to treat Robin boundary conditions in a posteriori error analysis on isotropic and anisotropic meshes. A numerical example supports the anisotropic error analysis. Mathematics Subject Classification (1991): 65N15, 65N30

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We report on the fabrication of pseudomorphic wurtzite Ga1−xMnxN grown on GaN with Mn concentrations up to 10% using molecular beam epitaxy. According to Rutherford backscattering, the Mn ions are mainly at the Ga-substitutional positions, and they are homogeneously distributed according to depth-resolved Auger-electron spectroscopy and secondary-ion mass-spectroscopy measurements. A random Mn distribution is indicated by transmission electron microscopy, and no Mn-rich clusters are present for optimized growth conditions. A linear increase of the c-lattice parameter with increasing Mn concentration is found using x-ray diffraction. The ferromagnetic behavior is confirmed by superconducting quantum-interference measurements showing saturation magnetizations of up to 150 emu/cm3.

Both for the H 1 -and L 2 -norms, we prove that, up to higher order perturbation terms, edge residuals yield global upper and local lower bounds on the error of linear finite element methods on anisotropic triangular or tetrahedral meshes. We also show that, with a correct scaling, edge residuals yield a robust error estimator for a singularly perturbed reaction-diffusion equation.Mathematics Subject Classification (1991): 65N30, 65N15

The paper deals with a singularly perturbed reaction diffusion model problem. The focus is on reliable a posteriori error estimators for the H 1 seminorm that can be applied to anisotropic finite element meshes. A residual error estimator and a local problem error estimator are proposed and rigorously analysed. They are locally equivalent, and both bound the error reliably. Furthermore three modifications of these estimators are introduced and discussed. Numerical experiments for all estimators complement and confirm the theoretical results.

A range of high quality Ga 1−x Mn x N layers have been grown by molecular beam epitaxy with manganese concentration 0.2  x  10%, having the x value tuned by changing the growth temperature (T g ) between 700 and 590 °C, respectively. We present a systematic structural and microstructure characterization by atomic force microscopy, secondary ion mass spectrometry, transmission electron microscopy, powder-like and high resolution X-ray diffraction, which do not reveal any crystallographic phase separation, clusters or nanocrystals, even at the lowest T g . Our synchrotron based X-ray absorption near-edge spectroscopy supported by density functional theory modelling and superconducting quantum interference device magnetometry results point to the predominantly +3 configuration of Mn in GaN and thus the ferromagnetic phase has been observed in layers with x > 5% at 3 < T < 10 K. The main detrimental effect of T g reduced to 590 o C is formation of flat hillocks, which increase the surface root-mean-square roughness, but only to mere 3.3 nm. Fine substrates' surface temperature mapping has shown that the magnitudes of both x and Curie temperature (T C ) correlate with local T g . It has been found that a typical 10 o C variation of T g across 1 inch substrate can lead to 40% dispersion of T C . The established here strong sensitivity of T C on T g turns magnetic measurements into a very efficient tool providing additional information on local T g , an indispensable piece of information for growth mastering of ternary compounds in which metal species differ in almost every aspect of their growth related parameters determining the kinetics of the growth. We also show that the precise determination of T C by two different methods, each sensitive to different moments of T C distribution, may serve as a tool for quantification of spin homogeneity within the material.

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