Purely algebraic domain decomposition methods for the incompressible Navier-Stokes equations

In the context of non overlapping domain decomposition methods, several algebraic approximations of the Dirichlet-to-Neumann (DtN) map are proposed in [F. X. Roux, et. al. Algebraic approximation of Dirichlet- to-Neumann maps for the equations of linear elasticity, Comput. Methods Appl. Mech. Engrg., 195, 2006, 3742-3759]. For the case of non overlapping domains, approximation to the DtN are analogous to the approximation of the Schur complements in the incomplete multilevel block factorization. In this work, several original and purely algebraic (based on graph of the matrix) domain decomposition techniques are investigated for steady state incompressible Navier-Stokes equation defined on uniform and stretched grid for low viscosity. Moreover, the methods proposed are highly parallel during both setup and application phase. Spectral and numerical analysis of the methods are also presented.Comment: Introduction rewritten, Comparison with state-of-art methods added, figure on overlapping case added, Complete algorithms added to build and solve with the preconditioners, Tests with Reynold number 3000 added, some observations with block jacobi method in analysis sectio

Analysis of hierarchical SSOR for three dimensional isotropic model problem

In this paper, we study a hierarchical SSOR (HSSOR) method which could be used as a standalone method or as a smoother for a two-grid method. It is found that the method leads to faster convergence compared to more costly incomplete LU (ILU(0)) with no fill-in, the SSOR, and the Block SSOR method. Moreover, for a two-grid method, numerical experiments suggests that HSSOR can be a better replacement for SSOR smoother both having no storage requirements and have no construction costs. Using Fourier analysis, ex- pressions for the eigenvalues and the condition number of HSSOR preconditioned problem is derived for the three-dimensional isotropic model problem.Comment: 14 pages, under submission in Journal, no figures. arXiv admin note: text overlap with arXiv:1105.346

Multilayer Edge Molecular Devices Based on Plasma Oxidation of Photolithographically Defined Bottom Metal Electrode

A multilayer edge molecular electronics device (MEMED), which utilize the two metal electrodes of a metal-insulator-metal tunnel junction as the two electrical leads to molecular channels, can overcome the long standing fabrication challenges for developing futuristic molecular devices. However, producing ultrathin insulator is the most challenging step in MEMED fabrication. A simplified molecular device approach was developed by avoiding the need of depositing a new materiel on the bottom electrode for growing ultrathin insulator. This paper discuss the approach for MEMED's insulator growth by one-step oxidation of a tantalum (Ta) bottom electrode, in the pholithographically defined region; i.e. ultrathin tantalum oxide (TaOx) insulator was grown by oxidizing bottom metal electrode itself. Organometallic molecular clusters (OMCs) were bridged across 1-3 nm TaOx along the perimeter of a tunnel junction to establish the highly efficient molecular conduction channels. OMC transformed the asymmetric transport profile of TaOx based tunnel junction into symmetric one. A TaOx based tunnel junction with top ferromagnetic (NiFe) electrode exhibited the transient current suppression by several orders. Further studies will be needed to strengthen the current suppression phenomenon, and to realize the full potential of TaOx based multilayer edge molecular spintronics devices.Comment: 14 Pages, 8 figure

Molecule Induced Strong Exchange Coupling between Ferromagnetic Electrodes of a Magnetic Tunnel Junction

Multilayer edge molecular spintronics device (MEMSD) approach can produce novel logic and memory units for the computers. MEMSD are produced by bridging the molecular channels across the insulator, in the exposed edge region(s) of a magnetic tunnel junction (MTJ). The bridged molecular channels start serving as the dominant exchange coupling medium between the two ferromagnetic electrodes of a MTJ. Present study focus on the effect of molecule enhanced exchange coupling on the magnetic properties of the MTJ. This paper shows that organometallic molecular clusters (OMCs) strongly increased the magnetic coupling between the two ferromagnetic electrodes. SQUID magnetometer showed that OMCs transformed the typical hysteresis magnetization curve of a Co/NiFe/AlOx/NiFe MTJ into linear one. Ferromagnetic resonance studies showed that OMC bridges affected the two fundamental resonance peaks of the Co/NiFe/AlOx/NiFe MTJ. According to magnetic force microscopy, OMCs caused the disappearance of magnetic contrast from the Co/NiFe/AlOx/NiFe tunnel junction area. These three independent and complimentary experiments, suggested the development of extremely strong interlayer exchange coupling. This work delineated a practical route to control the exchange coupling between ferromagnetic electrodes. Ability to tailor magnetic coupling can lead to the development of molecule based quantum computation device architecture.Comment: 34 Pages, 16 Figures, 1 Tabl

Excitation of Solar Acoustic Oscillations

The stochastic excitation of solar oscillations due to turbulent convection is reviewed. A number of different observational results that provide test for solar p-mode excitation theories are described. I discuss how well the stochastic excitation theory does in explaining these observations. The location and properties of sources that excite solar p-modes are also described. Finally, I discuss why solar g-modes should be linearly stable, and estimate the surface velocity amplitudes of low degree g-modes assuming that they are stochastically excited by the turbulent convection in the sun.Comment: 19 pages, LaTeX, 7 figues. Invited review to appear in the IAU symposium # 181 "Sounding Solar and Stellar Interiors", eds. F.X. Schmider & J. Provos

An Optimal Block Diagonal Preconditioner for Heterogeneous Saddle Point Problems in Phase Separation

The phase separation processes are typically modeled by Cahn-Hilliard equations. This equation was originally introduced to model phase separation in binary alloys, where phase stands for concentration of different components in alloy. When the binary alloy under preparation is subjected to a rapid reduction in temperature below a critical temperature, it has been experimentally observed that the concentration changes from a mixed state to a visibly distinct spatially separated two phase for binary alloy. This rapid reduction in the temperature, the so-called "deep quench limit", is modeled effectively by obstacle potential. The discretization of Cahn-Hilliard equation with obstacle potential leads to a block $2 \times 2$ {\em non-linear} system, where the $(1,1)$ block has a non-linear and non-smooth term. Recently a globally convergent Newton Schur method was proposed for the non-linear Schur complement corresponding to this non-linear system. The proposed method is similar to an inexact active set method in the sense that the active sets are first approximately identified by solving a quadratic obstacle problem corresponding to the $(1,1)$ block of the block $2 \times 2$ system, and later solving a reduced linear system by annihilating the rows and columns corresponding to identified active sets. For solving the quadratic obstacle problem, various optimal multigrid like methods have been proposed. In this paper, we study a non-standard norm that is equivalent to applying a block diagonal preconditioner to the reduced linear systems. Numerical experiments confirm the optimality of the solver and convergence independent of problem parameters on sufficiently fine mesh.Comment: 2 figure

The Distribution of Burst Energy and Shock Parameters for Gamma-ray Bursts

We calculate the statistical distribution of observed afterglow flux, in some fixed observed frequency band, and at some fixed observer time after the explosion ($t_{obs}$) in two models - one where the explosion takes place in a uniform density medium and the other where the surrounding medium has a power-law stratification such as is expected for a stellar wind. For photon energies greater than about 500 electron-volt and t_{obs}\gta 10^3 sec the afterglow flux distribution functions for the uniform ISM and the wind models are nearly identical. We compare the width of the theoretical distribution with the observed x-ray afterglow flux and find that the FWHM of the distribution for energy in explosion and the fractional energy in electrons ($\epsilon_e$) are each less than about one order of magnitude and the FWHM for the electron energy index is 0.6 or less.Comment: 12 pages & 3 figure, submitted to ApJ on 12/20/199

Gamma-ray Burst Energetics

We estimate the fraction of the total energy in a Gamma-Ray Burst (GRB) that is radiated in photons during the main burst. Random internal collisions among different shells limit the efficiency for converting bulk kinetic energy to photons. About 1% of the energy of explosion is converted to radiation, in 10-1000 kev energy band in the observer frame, for long duration bursts (lasting 10s or more); the efficiency is significantly smaller for shorter duration bursts. Moreover, about 50% of the energy of the initial explosion could be lost to neutrinos during the early phase of the burst if the initial fireball temperature is about 10 Mev or greater. If isotropic, the total energy budget of the brightest GRBs is about $10^{55}$erg, a factor of more than 20 larger than previously estimated. Anisotropy of explosion, as evidenced in two GRBs, could reduce the energy requirement by a factor of 10-100. Putting these two effects together we find that the energy release in the most energetic bursts is about 10$^{54}$ erg.Comment: 10 pages & 1 figure, final version as will appear in ApJ lette

Preconditioners for Saddle Point Problems on Truncated Domains in Phase Separation Modelling

The discretization of Cahn-Hilliard equation with obstacle potential leads to a block 2 by 2 non-linear system, where the p1, 1q block has a non-linear and non-smooth term. Recently a globally convergent Newton Schur method was proposed for the non-linear Schur complement corresponding to this non-linear system. The solver may be seen as an inexact Uzawa method which has the falvour of an active set method in the sense that the active sets are first identified by solving a quadratic obstacle problem corresponding to the p1, 1q block of the block 2 by 2 nonlinear system, and a new decent direction is obtained after discarding the active set region. The problem becomes linear on nonactive set, and corresponds to solving a linear saddle point problem on truncated domains. For solving the quadratic obstacle problem, various optimal multigrid like methods have been proposed. In this paper solvers for the truncated saddle point problem is considered. Three preconditioners are considered, two of them have block diagonal structure, and the third one has block tridiagonal structure. One of the block diagonal preconditioners is obtained by adding certain scaling of stiffness and mass matrices, whereas, the remaining two involves Schur complement. Eigenvalue bound and condition number estimates are derived for the preconditioned untruncated problem. It is shown that the extreme eigenvalues of the preconditioned truncated system remain bounded by the extreme eigenvalues of the preconditioned untruncated system. Numerical experiments confirm the optimality of the solvers.Comment: arXiv admin note: text overlap with arXiv:1601.0323

On random coarsening and its applications

In this paper, we use the Poincare separation theorem for estimating the eigenvalues of the fine grid. We propose a randomized version of the algorithm where several different coarse grids are constructed thus leading to more comprehensive eigenvalue estimates. The proposed algorithm is suited for modern day multicore and distributed processing