Weighted BFBT Preconditioner for Stokes Flow Problems with Highly Heterogeneous Viscosity

We present a weighted BFBT approximation (w-BFBT) to the inverse Schur complement of a Stokes system with highly heterogeneous viscosity. When used as part of a Schur complement-based Stokes preconditioner, we observe robust fast convergence for Stokes problems with smooth but highly varying (up to 10 orders of magnitude) viscosities, optimal algorithmic scalability with respect to mesh refinement, and only a mild dependence on the polynomial order of high-order finite element discretizations ($Q_k \times P_{k-1}^{disc}$, order $k \ge 2$). For certain difficult problems, we demonstrate numerically that w-BFBT significantly improves Stokes solver convergence over the widely used inverse viscosity-weighted pressure mass matrix approximation of the Schur complement. In addition, we derive theoretical eigenvalue bounds to prove spectral equivalence of w-BFBT. Using detailed numerical experiments, we discuss modifications to w-BFBT at Dirichlet boundaries that decrease the number of iterations. The overall algorithmic performance of the Stokes solver is governed by the efficacy of w-BFBT as a Schur complement approximation and, in addition, by our parallel hybrid spectral-geometric-algebraic multigrid (HMG) method, which we use to approximate the inverses of the viscous block and variable-coefficient pressure Poisson operators within w-BFBT. Building on the scalability of HMG, our Stokes solver achieves a parallel efficiency of 90% while weak scaling over a more than 600-fold increase from 48 to all 30,000 cores of TACC's Lonestar 5 supercomputer.Comment: To appear in SIAM Journal on Scientific Computin

Quantum interference in coherent tunnelling through branched molecular junctions containing ferrocene centers

In our theoretical study where we combine a nonequilibrium Green's function (NEGF) approach with density functional theory (DFT) we investigate branched compounds containing ferrocene moieties in both branches which due to their metal centers are designed to allow for asymmetry induced by local charging. In these compounds the ferrocene moieties are connected to pyridyl anchor groups either directly or via acetylenic spacers in a meta-connection where we also compare our results with those obtained for the respective single-branched molecules with both meta- and para-connections between the metal center and the anchors. We find a destructive quantum interference (DQI) feature in the transmission function slightly below the lowest unoccupied molecular orbital (LUMO) which dominates the conductance even for the uncharged branched compound with spacer groups inserted. In an analysis based on mapping the structural characteristics of the range of molecules in our article onto tight-binding models, we identify the structural source of the DQI minimum as the through-space coupling between the pyridyl anchor groups. We also find that local charging on one of the branches only changes the conductance by about one order of magnitude which we explain in terms of the spatial distributions of the relevant molecular orbitals for the branched compounds.Comment: Accepted for publication in Physical Review B, 12 pages in printed version, 10 figure

Charge localisation on a redox-active single molecule junction and its influence on coherent electron transport

For adjusting the charging state of a molecular metal complex in the context of a density functional theory description of coherent electron transport through single molecule junctions, we correct for self interaction effects by fixing the charge on a counterion, which in our calculations mimics the effect of the gate in an electrochemical STM setup, with two competing methods, namely the generalized $\Delta$ SCF technique and screening with solvation shells. One would expect a transmission peak to be pinned at the Fermi energy for a nominal charge of +1 on the molecule in the junction but we find a more complex situation in this multicomponent system defined by the complex, the leads, the counterion and the solvent. In particular equilibrium charge transfer between the molecule and the leads plays an importanty role, which we investigate in dependence on the total external charge in the context of electronegativity theory

A density functional theory based direct comparison of coherent tunnelling and electron hopping in redox-active single molecule junctions

For defining the conductance of single molecule junctions with a redox functionality in an electrochemical cell, two conceptually different electron transport mechanisms, namely coherent tunnelling and vibrationally induced hopping compete with each other, where implicit parameters of the setup such as the length of the molecule and the applied gate voltage decide which mechanism is the dominant one. Although coherent tunnelling is most efficiently described within Landauer theory, while the common theoretical treatment of electron hopping is based on Marcus theory, both theories are adequate for the processes they describe without introducing accuracy limiting approximations. For a direct comparison, however, it has to be ensured that the crucial quantities obtained from electronic structure calculations, i.e. the transmission function T(E) in Landauer theory, and the transfer integral V, the reorganisation energy $\lambda$ and the driving force $\Delta G^0$ in Marcus theory, are derived from similar grounds as pointed out by Nitzan and co-workers in a series of publications. In this article our framework is a single particle picture, where we perform density functional theory calculations for the conductance corresponding to both transport mechanisms for junctions with the central molecule containing one, two or three Ruthenium centers, respectively, where we extrapolate our results in order to define the critical length for the transition point of the two regimes which we identify at 5.76 nm for this type of molecular wire. We also discuss trends in dependence on an electrochemically induced gate potential