A posteriori error estimates for elliptic problems in two and three space dimensions

Let $u \in H$ be the exact solution of a given selfadjoint elliptic boundary value problem, which is approximated by some $\tilde u \in \mathcal{S}$, $\mathcal{S}$ being a suitable finite-element space. Efficient and reliable a posteriors estimates of the error $\| {u - \tilde u} \|$, measuring the (local) quality of $\tilde u$, play a crucial role in termination criteria and in the adaptive refinement of the underlying mesh. A well-known class of error estimates can be derived systematically by localizing the discretized defect problem by using domain decomposition techniques. In this paper, we provide a guideline for the theoretical analysis of such error estimates. We further clarify the relation to other concepts. Our analysis leads to new error estimates, which are specially suited to three space dimensions. The theoretical results are illustrated by numerical computations

Trends in the magnetic properties of Fe, Co and Ni clusters and monolayers on Ir(111), Pt(111) and Au(111)

We present a detailed theoretical investigation on the magnetic properties of small single-layered Fe, Co and Ni clusters deposited on Ir(111), Pt(111) and Au(111). For this a fully relativistic {\em ab-initio} scheme based on density functional theory has been used. We analyse the element, size and geometry specific variations of the atomic magnetic moments and their mutual exchange interactions as well as the magnetic anisotropy energy in these systems. Our results show that the atomic spin magnetic moments in the Fe and Co clusters decrease almost linearly with coordination on all three substrates, while the corresponding orbital magnetic moments appear to be much more sensitive to the local atomic environment. The isotropic exchange interaction among the cluster atoms is always very strong for Fe and Co exceeding the values for bulk bcc Fe and hcp Co, whereas the anisotropic Dzyaloshinski-Moriya interaction is in general one or two orders of magnitude smaller when compared to the isotropic one. For the magnetic properties of Ni clusters the magnetic properties can show quite a different behaviour and we find in this case a strong tendency towards noncollinear magnetism

An Explicit and Symplectic Integrator for Quantum-Classical Molecular Dynamics

An explicit and symplectic integrator called PICKABACK for quantum-classical molecular dynamics is presented. The integration scheme is time reversible and unitary in the quantum part. We use the Lie formalism in order to construct a formal evolution operator which is split by the Strang splitting yielding the symplectic discretization PICKABACK. Finally the new method is compared with a widely used hybrid method in two examples: a collinear collision of a particle with a quantum oscillator and, additionally, a photodissociation process of an ArHCl molecule. It is shown that the PICKABACK algorithm is more stable and accurate at no additional numerical effort

Isotope effects and possible pairing mechanism in optimally doped cuprate superconductors

We have studied the oxygen-isotope effects on T_{c} and in-plane penetration depth \lambda_{ab}(0) in an optimally doped 3-layer cuprate Bi_{1.6}Pb_{0.4}Sr_{2}Ca_{2}Cu_{3}O_{10+y} (T_{c} \sim 107 K). We find a small oxygen-isotope effect on T_{c} (\alpha_{O} = 0.019), and a substantial effect on \lambda_{ab} (0) (\Delta \lambda_{ab} (0)/\lambda_{ab} (0) = 2.5\pm0.5%). The present results along with the previously observed isotope effects in single-layer and double-layer cuprates indicate that the isotope exponent \alpha_{O} in optimally doped cuprates is small while the isotope effect on the in-plane effective supercarrier mass is substantial and nearly independent of the number of the CuO_{2} layers. A plausible pairing mechanism is proposed to explain the isotope effects, high-T_{c} superconductivity and tunneling spectra in a consistent way.Comment: 5 pages, 4 figure

Large oxygen-isotope effect in Sr_{0.4}K_{0.6}BiO_{3}: Evidence for phonon-mediated superconductivity

Oxygen-isotope effect has been investigated in a recently discovered superconductor Sr_{0.4}K_{0.6}BiO_{3}. This compound has a distorted perovskite structure and becomes superconducting at about 12 K. Upon replacing ^{16}O with ^{18}O by 60-80%, the T_c of the sample is shifted down by 0.32-0.50 K, corresponding to an isotope exponent of alpha_{O} = 0.40(5). This isotope exponent is very close to that for a similar bismuthate superconductor Ba_{1-x}K_{x}BiO_{3} with T_c = 30 K. The very distinctive doping and T_c dependencies of alpha_{O} observed in bismuthates and cuprates suggest that bismuthates should belong to conventional phonon-mediated superconductors while cuprates might be unconventional supercondutors.Comment: 9 pages, 5 figure