Poisson Equation

In this note, we propose and analyse a method for handling interfaces between nonmatching grids based on an approach suggested by Nitsche (1971) for the approximation of Dirichlet boundary conditions. The exposition is limited to self-adjoint elliptic problems, using Poisson's equation as a model. A priori and a posteriori error estimates are given. Some numerical results are included.

This paper describes our on-going work to accelerate ZENO, a software tool based on Monte Carlo methods (MCMs), used for computing material properties at nanoscale. ZENO employs three main algorithms: (1) Walk on Spheres (WoS), (2) interior sampling, and (3) surface sampling. We have accelerated the first two algorithms. For the sake of brevity, the paper will discuss our work on the first one only as it is the most commonly used and the acceleration techniques were similar in both cases. WoS is a Brownian motion MCM for solving a class of partial differential equations (PDEs). It provides a stochastic solution to a PDE by estimating the probability that a random walk, which started at infinity, will hit the surface of the material under consideration. WoS is highly effective when the problem's geometry is additive, as this greatly reduces the number of walk steps needed to achieve accurate results. The walks start on the surface of an enclosing sphere and can make much larger jumps than in a direct simulation of Brownian motion. Our current implementation represents the molecular structure of nanomaterials as a union of possibly overlapping spheres. The core processing bottleneck in WoS is a Computational Geometry one, as the algorithm repeatedly determines the distance from query point to the material surface in each step of the random walk. In this paper, we present results from benchmarking spatial data structures, including several open-source implementations of k-D trees, for accelerating WoS algorithmically. The paper also presents results from our multicore and cluster parallel implementation to show that it exhibits linear strong scaling with the number of cores and compute nodes; this implementation delivers up to 4 orders of magnitude speedup compared to the original FORTRAN code when run on 8 nodes (each with dual 6-core Intel Xeon CPUs) with 24 threads per node.

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A three-dimensional simulation approach for the investigation of Carbon NanoTube Field Effect Transistors (CNTFETs) is presented, based on the self-consistent solution of the 3D Poisson-Schrödinger equation with open boundary conditions, within the Non-Equilibrium Green's Function formalism. This approach enables accurate simulations of transport through ballistic CNTFETs for generic device structures, eliminating the use of ideal geometries, such as coaxial devices, often used in order to simplify the electrostatics of the device.

Techniques to engineer a MOSFET's channel in the lateral direction have been proposed to enhance the device performance. In this paper, we present a thorough simulation study to evaluate the feasibility of such lateral engineering techniques. Each of three types of transport equations, the ballistic Boltzmann, drift-diffusion and non-equilibrium Green's function with scattering, is solved self-consistently with 2-D Poisson equation to simulate device performance under both the ballistic and dissipative transport conditions. The results indicate that even if highly idealized device structures are assumed, only limited improvements over the conventional MOSFETs can be achieved by the channel engineering techniques. These results don't conflict with reports of large on-current improvements using the lateral channel engineering, because those comparisons with the conventional MOSFETs were done without specifying a common off-current.

We present numerical schemes for the incompressible Navier-Stokes equations based on a primitive variable formulation in which the incompressibility constraint has been replaced by a pressure Poisson equation. The pressure is treated explicitly in time, completely decoupling the computation of the momentum and kinematic equations. The result is a class of extremely efficient Navier-Stokes solvers. Full time accuracy is achieved for all flow variables. The key to the schemes is a Neumann boundary condition for the pressure Poisson equation which enforces the incompressibility condition for the velocity field. Irrespective of explicit or implicit time discretization of the viscous term in the momentum equation the explicit time discretization of the pressure term does not affect the time step constraint. Indeed, we prove unconditional stability of the new formulation for the Stokes equation with explicit treatment of the pressure term and first or second order implicit treatment of the viscous term. Systematic numerical experiments for the full Navier-Stokes equations indicate that a second order implicit time discretization of the viscous term, with the pressure and convective terms treated explicitly, is stable under the standard CFL condition. Additionally, various numerical examples are presented, including both implicit and explicit time discretizations, using spectral and finite difference spatial discretizations, demonstrating the accuracy, flexibility and efficiency of this class of schemes. In particular, a Galerkin formulation is presented requiring only C 0 elements to implement.

Carbon nanotube field-effect transistors (CNTFETs) have been studied in recent years as a potential alternative to CMOS devices, because of the capability of ballistic transport. The ambipolar behavior of Schottky barrier CNTFETs limits the performance of these devices. A double gate design is proposed to suppress this behavior. In this structure the first gate located near the source contact controls carrier injection and the second gate located near the drain contact suppresses parasitic carrier injection. To avoid the ambipolar behavior it is necessary that the voltage of the second gate is higher or at least equal to the drain voltage. The behavior of these devices has been studied by solving the coupled Schrödinger-Poisson equation system. We investigated the effect of the second gate voltage on the performance of the device and finally the advantages and disadvantages of these options are discussed.

In this paper, we propose a multi-grid algorithm for solving the Navier-Stokes equations and we analyze its convergence. The proposed method consists of multi-level grids, where in the first one (the very coarse mesh ℎ 0 = 𝐻) we solve one small nonlinear system of discrete Navier-Stokes equations, then, in every intermediate mesh (ℎ 𝑖 = ℎ 2 𝑖-1 = 𝐻 2 𝑖 ) we solve two linear problems and in the last one (the very fine mesh ℎ𝑛 = 𝐻 2 𝑛 ) we solve one linear system. Moreover, the algorithm produces a numerical solution with an optimal asymptotic error estimate with respect to ℎ = ℎ𝑛. Finally, we give some numerical illustrations showing the efficiency of the new multi-grid algorithm.

We present preliminary results on a new family of distribution functions that are able to generate axisymmetric, truncated (i.e., finite size) stellar dynamical models characterized by toroidal shapes. The relevant distribution functions generalize those that are known to describe polytropic spheres, for which all the dynamical and structural properties of the system can be expressed in explicit form as elementary functions of the system gravitational potential. The model construction is then completed by a numerical study of the associated Poisson equation. We note that our axisymmetric models can also include the presence of an external gravitational field, such as that produced by a massive disk or by a central mass concentration (e.g., a supermassive black hole).

Context. That the rotation curves of spiral galaxies are generally flat in the outer regions is commonly considered an indication that galaxy disks are embedded in quasi-isothermal halos. In practice, disk-halo decompositions of galaxy rotation curves are performed in a parametric way by modeling the halo force contribution by means of expressions that approximately describe the properties of the regular isothermal sphere or other spherical density distributions suggested by cosmological simulations. Aims. In this paper we construct self-consistent models of nonspherical isothermal halos embedding a zero-thickness disk, by assuming that the halo distribution function is a Maxwellian. The general method developed here can also be used to study the properties of other physically-based choices for the halo distribution function and to the case of a disk accompanied by a bulge. Methods. The construction was performed by means of an iterative procedure that generalizes a method introduced in the past to construct spheroidal models of rotating elliptical galaxies. In a preliminary investigation, which set the empirical framework to study the self-consistent models developed in this paper, we note the existence of a fine tuning between the scalelengths R Ω and h, respectively characterizing the rise of the rotation curve and the luminosity profile of the disk, which surprisingly applies to both high surface brightness and low surface brightness galaxies in similar ways. We show that this empirical correlation identifies a much stronger conspiracy than the one required by the smoothness and flatness of the rotation curve and often referred to as disk-halo conspiracy. Results. As a natural property, the self-consistent models presented in this paper are found to be characterized by smooth and flat rotation curves for very different disk-to-halo mass ratios, hence suggesting that conspiracy is not as dramatic as often imagined. For a typical, observed rotation curve, with asymptotically flat rotation curve at V ∞ (the precise value of which can also be treated as a free parameter), and a typical density profile of the disk, the relevant self-consistent models are characterized by two dimensionless parameters, which correspond to the dimensional scales (the disk mass-to-light ratio M/L and the halo central density) of standard disk-halo decompositions. We find that, if the rotation curve is decomposed by means of our self-consistent models, the disk-halo degeneracy is removed and that typical rotation curves are fitted by models that are below the maximum-disk prescription. Similar results are obtained from a study of NGC 3198. Finally, we quantify the flattening of the spheroidal halo, which is significant, especially on the scale of the visible disk.

Modified Newtonian Dynamics (MOND) is a theory that modifies Newton's force law to explain observations that most astronomers interpret as evidence for dark matter. Observations of 1E 0657-558, two colliding galaxy clusters known colloquially as the "Bullet Cluster," offer the best evidence against MOND and MOND-like theories and some of the best evidence for dark matter. We present herein an overview of the theory of MOND and its success at explaining certain observations as well as the Bullet Cluster and its case against MOND. Contents 1 Modified Newtonian Dynamics (MOND) 3

The axisymmetric turbulent far wake of a long slender cylinder is being studied. Classical analytical solutions describing the wake width and the velocity defect are presented. The governing equations have been solved with appropriate boundary conditions and a scaling analysis to obtain a similarity equation for the mean velocity distributions. Solutions of the velocity distributions are presented in terms of the hypergeometric function known as the Whittaker function. Validation of the analysis is done with available experimental data.

Monte Carlo simulations coupled self-consistently with the three-dimensional Poisson equation are carried out under the double-gate MOSFET structures. The Coulomb force experienced by an electron inside the device is directly evaluated by performing the Monte Carlo simulations with or without the full Coulomb interaction and the Coulomb force on the channel electron corresponding to plasmon excitations is clarified. It is pointed out that the consistency of the boundary condition is achieved only if the long-range Coulomb interaction is properly taken into account, and this is crucial for predicting reliable device characteristics in ultra-small devices. The drain current and transconductance are greatly degraded if the self-consistent potential fluctuations are taken into account.

In Gresho and Sani (Int. J. Numer. Methods Fluids 1987; 7:1111–1145; Incompressible Flow and the Finite Element Method, vol. 2. Wiley: New York, 2000) was proposed an important hypothesis regarding the pressure Poisson equation (PPE) for incompressible flow: Stated there but not proven was a so‐called equivalence theorem (assertion) that stated/asserted that if the Navier–Stokes momentum equation is solved simultaneously with the PPE whose boundary condition (BC) is the Neumann condition obtained by applying the normal component of the momentum equation on the boundary on which the normal component of velocity is specified as a Dirichlet BC, the solution (u, p) would be exactly the same as if the ‘primitive’ equations, in which the PPE plus Neumann BC is replaced by the usual divergence‐free constraint (∇ · u = 0), were solved instead.This issue is explored in sufficient detail in this paper so as to actually prove the theorem for at least some situations. Additionally, like the ori...

In this work, the long-time behavior of the solutions of the Vlasov–Poisson system when starting near a spatially homogeneous equilibrium is analyzed numerically. It is found that the asymptotic state toward which the system evolves is not a Bernstein–Green–Kruskal (BGK) equilibrium, but can rather be described by a superposition of BGK modes. Also, it is shown that the small oscillations at the plasma frequency after saturation in the one-sided bump-on-tail instability are due to beating between the unstable mode and the least stable mode.

In this work we analyze the existence and regularity of the solution of a nonhomogeneous Neumann problem for the Poisson equation in a plane domain Ω with an external cusp. In order to prove that there exists a unique solution in H 1 (Ω) using the Lax-Milgram theorem we need to apply a trace theorem. Since Ω is not a Lipschitz domain, the standard trace theorem for H 1 (Ω) does not apply, in fact the restriction of H 1 (Ω) functions is not necessarily in L 2 (∂Ω). So, we introduce a trace theorem by using weighted Sobolev norms in Ω. Under appropriate assumptions we prove that the solution of our problem is in H 2 (Ω) and we obtain an a priori estimate for the second derivatives of the solution.

The effect of space charge rotation has been studied in Vlasov-Poisson potential function with KV distribution in quadrupole. Effective potential varies due to rotation and the related parameter has been obtained based on focusing and defocusing of beam particles. The changed beam plasma frequencies have been calculated in the frame of laboratory coordinates. Also the new Hamiltonian of the system has been derived and finally the equipotential lines, electric and magnetic fields have been obtained. The magnetic field remains without any changing, but the electric field and equipotential lines changed in static qudrupole. The result of rotational space charge in static quadrupole is comparable with rotation of quadrupole system.

The dispersion equations for bulk and local phonon-plasmon (PP) waves in layered conductors and superlattices are obtained in the frame of a model which neglects electron hopping between layers but assumes an arbitrary charge-charge response function within the layers. The local modes propagate in the defect layer which differs from the rest of the layers by density and mass of charge carriers. The response function is found in the model of a two-dimensional electron gas interacting with acoustic lattice vibrations in the long wavelength approximation. It is shown that in this model two bulk PP waves propagates within the narrow low-frequency and broad high-frequency bands and three local modes exist beyond these bands. Depending on the sign and magnitude of the defect layer parameter : (which is a relative variation of the ratio: density to mass of charge carriers) two acoustic and one optic local PP modes exist in the system.

In the present paper, a two-dimensional (2-D) analytical model for graded channel fully depleted cylindrical/surrounding gate MOS-FET (GC FD CGT/SGT) has been developed by solving the Poisson's equation in cylindrical coordinates. An abrupt transition of silicon film doping at the interface has been assumed and the effects of the doping and the lengths of the high and low doped regions have been taken into account. The model is used to obtain the expressions of surface potential and electric field in the two regions. The analysis is extended to obtain the expressions for threshold voltage (V th ) and subthreshold swing. It is shown that a graded doping profile in the channel leads to suppression of short channel effects (SCEs) like threshold voltage roll-off, drain induced barrier lowering (DIBL) and hot carrier effects. The results so obtained have been compared with simulated results obtained using the device simulator ATLAS 3D and are found to be in good agreement.

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We present a fast Poisson solver on spherical shells. With a special change of variable, the radial part of the Laplacian transforms to a constant coefficient differential operator. As a result, the Fast Fourier Transform can be applied to solve the Poisson equation with O(N 3 log N) operations. Numerical examples have confirmed the accuracy and robustness of the new scheme.

We present a fast Poisson solver on spherical shells. With a special change of variable, the radial part of the Laplacian transforms to a constant coefficient differential operator. As a result, the Fast Fourier Transform can be applied to solve the Poisson equation with operations. Numerical examples have confirmed the accuracy and robustness of the new scheme.

The Shortest-Remaining-Processing-Time (SRPT) scheduling policy has long been known to be optimal for minimizing mean response time (sojourn time). Despite this fact, SRPT scheduling is rarely used in practice. It is believed that the performance improvements of SRPT over other scheduling policies stem from the fact that SRPT unfairly penalizes the large jobs in order to help the small jobs. This belief has led people to instead adopt "fair" scheduling policies such as Processor-Sharing (PS), which produces the same expected slowdown for jobs of all sizes. This paper investigates formally the problem of unfairness in SRPT scheduling as compared with PS scheduling. The analysis assumes an M/G/1 model, and emphasizes job size distributions with a heavy-tailed property, as are characteristic of empirical workloads. The analysis shows that the degree of unfairness under SRPT is surprisingly small. The M/G/1/SRPT and M/G/1/PS queues are also analyzed under overload and closed-form expressions for mean response time as a function of job size are proved in this setting. £

We present a general mesh-free description of the magnetic field distribution in various electromagnetic machines, actuators, and devices. Our method is based on transfer relations and Fourier theory, which gives the magnetic field solution for a wide class of twodimensional (2-D) boundary value problems. This technique can be applied to rotary, linear, and tubular permanent-magnet actuators, either with a slotless or slotted armature. In addition to permanent-magnet machines, this technique can be applied to any 2-D geometry with the restriction that the geometry should consist of rectangular regions. The method obtains the electromagnetic field distribution by solving the Laplace and Poisson equations for every region, together with a set of boundary conditions. Here, we compare the method with finite-element analyses for various examples and show its applicability to a wide class of geometries. Index Terms-Boundary value problem, Fourier analysis, permanent magnet. LIST OF SYMBOLS Magnetic vector potential m Magnetic flux density vector Remanent flux density Unit vector-Magnetic field strength vector m Height Current density vector Region number-Longitudinal direction Magnetization vector m Harmonic number Harmonic number Normal direction Tangential direction Spatial frequency or rad Magnetic susceptibility Offset in tangential direction Permeability m Permeability of vacuum m Relative permeability Angular direction Width or rad Bessel function of first kind of 0th order Bessel function of first kind of 1th order Bessel function of second kind of 0th order Bessel function of second kind of 1th order Manuscript

We establish the vanishing viscosity limit of the Navier-Stokes equations to the isentropic Euler equations for one-dimensional compressible fluid flow. For the Navier-Stokes equations, there exist no natural invariant regions for the equations with the real physical viscosity term so that the uniform sup-norm of solutions with respect to the physical viscosity coefficient may not be directly controllable and, furthermore, convex entropy-entropy flux pairs may not produce signed entropy dissipation measures. To overcome these difficulties, we first develop uniform energy-type estimates with respect to the viscosity coefficient for the solutions of the Navier-Stokes equations and establish the existence of measure-valued solutions of the isentropic Euler equations generated by the Navier-Stokes equations. Based on the uniform energy-type estimates and the features of the isentropic Euler equations, we establish that the entropy dissipation measures of the solutions of the Navier-Stokes equations for weak entropy-entropy flux pairs, generated by compactly supported C 2 test functions, are confined in a compact set in H −1 , which lead to the existence of measure-valued solutions that are confined by the Tartar-Murat commutator relation. A careful characterization of the unbounded support of the measure-valued solution confined by the commutator relation yields the reduction of the measure-valued solution to a Delta mass, which leads to the convergence of solutions of the Navier-Stokes equations to a finite-energy entropy solution of the isentropic Euler equations.

A kinetic theory of the Debye sheath in the Tonks–Langmuir model of the plasma-wall transition layer with hot neutrals is presented. The plasma, consisting of Boltzmann-distributed electrons and singly charged ions, is in contact with an absorbing negative wall. Dependencies of electric-potential characteristics on neutrals temperature are investigated for the first time. The results can be generalized to other sheath models.

An adaptive mesh method is presented for numerical simulation of shock-induced combustion. The method is composed of two independent ingredients: a flow solver and a mesh redistribution algorithm. The flow solver is a finite-volume based second-order upwind TVD scheme, together with a lower-upper symmetric Gauss-Seidel relaxation scheme for solving the multispecies Navier-Stokes equations with finite rate chemistry. The adaptive mesh is determined by a grid generation method based on solving Poisson equations, with the monitor function carefully designed to resolve both sharp fronts and the induction zone between them. Numerically it is found that the resolution of the induction zone is particularly critical to the combustion problems, and an under-resolved numerical method may cause excessive energy release and spurious runaway reactions. The monitor function proposed in this paper, which is based on the relative rate of change of mass fraction, covered this issue satisfactorily. Numerical simulations of supersonic flows past an axisymmetric projectile in a premixed hydrogen/oxygen (or air) mixture are carried out. The results show that the spurious runaway chemical reactions appearing on coarse grids can be eliminated by using adaptive meshes without invoking any ad hoc treatment for reaction rates. The adaptive mesh approach is more effective than the fixed mesh one in obtaining grid-independent results. Finally, discrepancies between the numerical and benchmark experimental results and difficulties in simulating the detonation waves are delineated which appeal for further investigation.

Figure 1: A range image set and its parametrization. A set of range images U i is captured by direct measurement of real world objects or by means of digital rendering of virtual objects. Each range scan U i is mapped over a domain D i by a parametrization function c i , in a globally consistent way. Among other uses, a semi-regular quad remeshing of the original object can be obtained by regularly sampling ∪D i .

A highly phosphorus δ-doped Si device is modeled with a quantum well with periodic boundary conditions and the semi-empirical spds* tight-binding band model. Its temperaturedependent electronic properties are studied. To account for high doping density with many electrons, a highly parallelized selfconsistent Schrödinger-Poisson solver is used with atomistic representations of multiple impurity ions. The band-structure in equilibrium and the corresponding Fermi-level position are computed for a selective set of temperatures. The result at room temperature is compared with previous studies and the temperature-dependent electronic properties are discussed further in detail with the calculated 3-D self-consistent potential profile.

A two-phase flow extended finite element method (XFEM) model is presented to analyze the injection and sequestration of carbon dioxide (CO2) in deep saline aquifers. Carbon sequestration is a multiscale problem, involving length scales over four orders of magnitude–from injection well diameter scale to the aquifer scale. XFEM is introduced to accurately approximate near injection well pressure behaviour with elements significantly larger than the injection well diameter. We present a vertically averaged multiphase flow model that combines XFEM to approximate the pressure field, with a Streamline Upwind Petrov-Galerkin / Finite Element / Finite Difference method (SUPG-FEM-FDM) to approximate the distribution of CO2 in the aquifer. Near well enrichment functions are presented along with the solution procedure for the coupled problem. Two examples are presented; in the first, CO2 injection into a perfectly horizontal aquifer is model with both XFEM and FEM-based methods. It is shown th...

The solar wind interacts with planetary magnetospheres, generating plasma waves in both the upstream region and the magnetospheric environment. Mars, lacking an inherent magnetic field, has an induced magnetosphere formed through solar wind interaction with its ionosphere. These waves are crucial for momentum and energy exchange within the planetary plasma environment. This study focuses on the existence and dynamics of electrostatic solitary waves (ESWs) in Martian ionospheric plasma, characterized by various flowing ions. We employ a multifluid plasma model incorporating positive (streaming) ions, negative ions, and two distinct kappa-distributed electron populations to study ESW propagation in the Martian ionosphere from first principles. Linear analysis reveals four distinct modes, including a subsonic mode due to the negative-ion beam. Electrostatic waves may become unstable due to a beam instability excited at long wavelengths. Seeking stationary profile solutions, an energy balance equation is obtained in the moving reference frame, and the shape of the solitary wave can thus be predicted numerically. A meticulous analysis reveals that either positive-or negative-polarity ESWs (or both) may occur (simultaneously) in the Martian environment. In addition to conventional bipolar E-field waveforms, our theoretical model predicts the existence of wiggly bipolar pulses (supersolitary waves) and offset bipolar pulses (flat-top solitary waves) in Martian plasma. Comparison of our model's predictions with real observational plasma parameters indicates that, like Martian magnetosheath plasma, ionospheric plasma may sustain ESWs measuring several tens of millivolts per meter.

We analyze electrostatic deformations of rectangular, annular circular, solid circular, and elliptic micro-electromechanical systems (MEMS) by modeling them as elastic membranes. The nonlinear Poisson equation governing their deformations is solved numerically by the meshless local Petrov-Galerkin (MLPG) method. A local symmetric augmented weak formulation of the problem is introduced, and essential boundary conditions are enforced by introducing a set of Lagrange multipliers. The trial functions are constructed by using the moving least-squares approximation, and the test functions are chosen from a space of functions different from the space of trial solutions. The resulting nonlinear system of equations is solved by using the pseudoarclength continuation method. Presently computed values of the pull-in voltage and the maximum pull-in deflection for the rectangular and the circular MEMS are found to agree very well with those available in the literature; results for the elliptic MEMS are new.

This paper concerns an optimization problem related to the Poisson equation for the p-Laplace operator, subject to homogeneous Dirichlet boundary conditions. Physically the Poisson equation models, for example, the deformation of a nonlinear elastic membrane which is fixed along the boundary, under load. A particular situation where the load is represented by a characteristic function is investigated.

Submitted for the DFD05 Meeting of The American Physical Society Electroosmotic flow in rectangular microchannels: numerical simulation and asymptotic theory 1 SUBHRA DATTA, SANDIP GHOSAL, NEE-LESH PATANKAR, Northwestern University-The problem of fluid flow in a microfluidic channel of rectangular cross-section is solved numerically when the zeta potential is not uniform. Variations in the axial direction as well as along the perimeter of the channel cross-section is considered. Excellent agreement is found with a previously published (Ghosal, 2002 JFM vol.459 pg. 103) asymptotic theory based on the lubrication approximation, even when the length scale of axial variations is of the same order as the characteristic channel width.

In questa appendice richiameremo e dimostreremo alcuni risultati di base sulle funzioni olomorfe a valori vettoriali. A tal fine ricordiamo in primis la definizione di integrale per una funzione continua a valori vettoriali, rinviando a [6, Ch.3], per ulteriori approfondimenti. Definizione A.1. Siano (X, •) uno spazio di Banach e sia f : [a, b] → X una funzione limitata. Diciamo che fè integrabile su [a, b] se esiste x ∈ X tale che per ogni ε > 0 esiste δ > 0 tale che per ogni partizione {a = t 0 < t 1 <. .. < t n = b} di [a, b] con sup i=1,...,n (t i − t i−1) < δ e per ogni scelta di punti ξ i ∈ [t i−1 , t i ] risulta 65 brought to you by CORE View metadata, citation and similar papers at core.ac.uk

insieme non vuoto e A un sottospazio vettoriale dello spazio delle funzioni a valori reali (risp. complessi) su X. Si dice che Aè un algebra funzionale reale (risp. complessa) se per ogni f, g ∈ A si ha che f g ∈ A. Un sottospazio B ⊆ Aè detto una sottoalgebra di A se Bè un algebra. Esempio B.2. Sia X uno spazio topologico e sia C(X, R) lo spazio delle funzioni reali e continue su X. C(X, R)è un'algebra reale funzionale. Il sottospazio BC(X, R) delle funzioni reali limitate e continue su Xè una sottoalgebra di C(X, R). E noto che lo spazio BC(X, R) dotato della norma f ∞ := sup x∈X |f (x)| (f ∈ BC(X, R)) brought to you by CORE View metadata, citation and similar papers at core.ac.uk

This paper describes a numerical experiment of magnetization switching driven by spin-polarized current in high-TMR magnetic tunnel junctions (TMR>100%). Differently from other works, the current density distribution throughout the cross-sectional area of the free-layer is here computed dinamically, by modeling the ferromagnet/insulator/ferromagnet trilayer as a series of parallel resistances. The validity of the main postulated hypothesis, which states that the current density vector is perpendicular to the sample plane, has been verified by numerically solving the Poisson equation. Our results show that the nonuniform current density distribution is a source of asymmetry for the switching. Furthermore, we found out that the switching processes are characterized by well defined localized pre-switching oscillation modes.

A new grid-less (no collocation) spectral projection method is presented. The unsteady Navier-Stokes equations are approximated according to the variational framework of Guermond and Quartapelle which accommodates two vector spaces for the velocity fields obtained in the two half-steps of the fractional-step method but retains only one in the final solution algorithm. Two different bases built on Legendre polynomials are used for the velocity and pressure to solve the corresponding Helmholtz and Poisson equations by direct spectral elliptic solvers. Interpolations P N and P N −2 are employed for velocity and pressure to satisfy the LBB stability requirement and a Gauss-Legendre quadrature formula with 3 2 N integration points is used to prevent aliasing error in the pseudospectral evaluation of the nonlinear terms. A BDF second-order time stepping is implemented to provide accurate numerical results about the stability of the singular driven cavity problem.

Pumping fluid is one of the crucial parts of any microfluidic system. Using electric and magnetic fields as a substitute for moving parts can have many advantages. In this study hydrodynamic and heat transfer characteristics of electroosmotic flow under influence of lateral electric and transverse magnetic field, are studied numerically. Results indicate that the dimensionless parameters such as Hartmann number, intensity of the lateral electric field, pressure gradient parameter and aspect ratio have an important role in controlling flow. It can be implied that the enhancement of pressure gradient leads to the decrease of critical Hartmann number, and this dependency can be reduced from 44% to 7% for S=0.5 to S=50 in two pressure gradients of Ω=1 and Ω=20. In addition, the reduction of aspect ratio of microchannel section leads to the increment of critical Hartmann number in a specified lateral electric field. At the end, thermal analysis is being done by consideration of the effects of magnetic and electric fields on the Nusselt number.

For a family of second-order elliptic operators with rapidly oscillating periodic coefficients, we study the asymptotic behavior of the Green and Neumann functions, using Dirichlet and Neumann correctors. As a result we obtain asymptotic expansions of Poisson kernels and the Dirichlet-to-Neumann maps as well as near optimal convergence rates in W 1,p for solutions with Dirichlet or Neumann boundary conditions.

To solve the elliptic boundary value problems with singularities, the simpli®ed hybrid combinations of the Ritz-Galerkin and ®nite element methods (RGM-FEM) are explored to lead to the global superconvergence rates on the entire solution domain, based on an a posteriori interpolation techniques of Lin and Yan [12] that only cost a little more computation. Let the solution domain S S 1 S 2 C 0 and S 1 S 2 Y. Suppose that S 1 can be partitioned into quasiuniform rectangles: S 1 ij ( ij , a singular point occurs at dS 2 , and the singular functions are chosen in S 2 . Then for bilinear elements, it is proven that the simpli®ed hybrid combinations of RGM-FEM can provide the global superconvergence rate Oh 2 for solution gradients over the entire subdomains S 1 and S 2 , where h is the maximal boundary length of ( ij . The global superconvergence Oh 2 is better, compared to Oh 2Àd , 0 `d ( 1 given in . Note that numerical stability of the simpli®ed hybrid combinations of RGM-FEM is also optimal . This paper presents the important results for the general case of Poisson-problems on a polygonal domain S estimates for the Sobolev norm k Á k 1 , given in a much more general sense than known before, cf. [1±4, 14±18].

To solve the elliptic boundary value problems with singularities, the simpli®ed hybrid combinations of the Ritz-Galerkin and ®nite element methods (RGM-FEM) are explored to lead to the global superconvergence rates on the entire solution domain, based on an a posteriori interpolation techniques of Lin and Yan [12] that only cost a little more computation. Let the solution domain S S 1 S 2 C 0 and S 1 S 2 Y. Suppose that S 1 can be partitioned into quasiuniform rectangles: S 1 ij ( ij , a singular point occurs at dS 2 , and the singular functions are chosen in S 2 . Then for bilinear elements, it is proven that the simpli®ed hybrid combinations of RGM-FEM can provide the global superconvergence rate Oh 2 for solution gradients over the entire subdomains S 1 and S 2 , where h is the maximal boundary length of ( ij . The global superconvergence Oh 2 is better, compared to Oh 2Àd , 0 `d ( 1 given in . Note that numerical stability of the simpli®ed hybrid combinations of RGM-FEM is also optimal . This paper presents the important results for the general case of Poisson-problems on a polygonal domain S estimates for the Sobolev norm k Á k 1 , given in a much more general sense than known before, cf. [1±4, 14±18].