Variational approximation of flux in conforming finite element methods for elliptic partial differential equations: a model problem

We consider the approximation of elliptic boundary value problems by conforming finite element methods. A model problem, the Poisson equation with Dirichlet boundary conditions, is used to examine the convergence behavior of flux defined on an internal boundary which splits the domain in two. A variational definition of flux, designed to satisfy local conservation laws, is shown to lead to improved rates of convergence

Existence and equilibration of global weak solutions to finitely extensible nonlinear bead-spring chain models for dilute polymers

We show the existence of global-in-time weak solutions to a general class of coupled FENE-type bead-spring chain models that arise from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The class of models involves the unsteady incompressible Navier-Stokes equations in a bounded domain in two or three space dimensions for the velocity and the pressure of the fluid, with an elastic extra-stress tensor appearing on the right-hand side in the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined by the Kramers expression through the associated probability density function that satisfies a Fokker-Planck-type parabolic equation, a crucial feature of which is the presence of a center-of-mass diffusion term. We require no structural assumptions on the drag term in the Fokker-Planck equation; in particular, the drag term need not be corotational. With a square-integrable and divergence-free initial velocity datum for the Navier-Stokes equation and a nonnegative initial probability density function for the Fokker-Planck equation, which has finite relative entropy with respect to the Maxwellian of the model, we prove the existence of a global-in-time weak solution to the coupled Navier-Stokes-Fokker-Planck system. It is also shown that in the absence of a body force, the weak solution decays exponentially in time to the equilibrium solution, at a rate that is independent of the choice of the initial datum and of the centre-of-mass diffusion coefficient.Comment: 75 page

Existence and equilibration of global weak solutions to Hookean-type bead-spring chain models for dilute polymers

We show the existence of global-in-time weak solutions to a general class of coupled Hookean-type bead-spring chain models that arise from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The class of models involves the unsteady incompressible Navier-Stokes equations in a bounded domain in two or three space dimensions for the velocity and the pressure of the fluid, with an elastic extra-stress tensor appearing on the right-hand side in the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined by the Kramers expression through the associated probability density function that satisfies a Fokker-Planck-type parabolic equation, a crucial feature of which is the presence of a center-of-mass diffusion term. We require no structural assumptions on the drag term in the Fokker-Planck equation; in particular, the drag term need not be corotational. With a square-integrable and divergence-free initial velocity datum for the Navier-Stokes equation and a nonnegative initial probability density function for the Fokker-Planck equation, which has finite relative entropy with respect to the Maxwellian of the model, we prove the existence of a global-in-time weak solution to the coupled Navier-Stokes-Fokker-Planck system. It is also shown that in the absence of a body force, the weak solution decays exponentially in time to the equilibrium solution, at a rate that is independent of the choice of the initial datum and of the centre-of-mass diffusion coefficient.Comment: 86 page

Analysis of a stabilised finite element method for power-law fluids

A low-order finite element method is constructed and analysed for an incompressible non-Newtonian flow problem with power-law rheology. The method is based on a continuous piecewise linear approximation of the velocity field and piecewise constant approximation of the pressure. Stabilisation, in the form of pressure jumps, is added to the formulation to compensate for the failure of the inf-sup condition, and using an appropriate lifting of the pressure jumps a divergence-free approximation to the velocity field is built and included in the discretisation of the convection term. This construction allows us to prove the convergence of the resulting finite element method for the entire range r > 2d of the power-law index r for which weak solutions to the model are known to exist in d space dimensions, d ∈ {2, 3}

Analysis of a stabilised finite element method for power-law fluids

A low-order finite element method is constructed and analysed for an incompressible non-Newtonian flow problem with power-law rheology. The method is based on a continuous piecewise linear approximation of the velocity field and piecewise constant approximation of the pressure. Stabilisation, in the form of pressure jumps, is added to the formulation to compensate for the failure of the inf-sup condition, and using an appropriate lifting of the pressure jumps a divergence-free approximation to the velocity field is built and included in the discretisation of the convection term. This construction allows us to prove the convergence of the resulting finite element method for the entire range r>2dd+2 of the power-law index r for which weak solutions to the model are known to exist in d space dimensions, d∈ { 2 , 3 }

Existence of global weak solutions to dumbbell models for dilute polymers with microscopic cut-off

Accepted versio

Four electrons in a two-leg Hubbard ladder: exact ground states

In the case of a two-leg Hubbard ladder we present a procedure which allows the exact deduction of the ground state for the four particle problem in arbitrary large lattice system, in a tractable manner, which involves only a reduced Hilbert space region containing the ground state. In the presented case, the method leads to nine analytic, linear, and coupled equations providing the ground state. The procedure which is applicable to few particle problems and other systems as well is based on an r-space representation of the wave functions and construction of symmetry adapted orthogonal basis wave vectors describing the Hilbert space region containing the ground state. Once the ground state is deduced, a complete quantum mechanical characterization of the studied state can be given. Since the analytic structure of the ground state becomes visible during the use of the method, its importance is not reduced only to the understanding of theoretical aspects connected to exact descriptions or potential numerical approximation scheme developments, but is relevant as well for a large number of potential technological application possibilities placed between nano-devices and quantum calculations, where the few particle behavior and deep understanding are important key aspects to know.Comment: 19 pages, 5 figure

Human T-Lymphotropic Virus-1 Visualized at the Virological Synapse by Electron Tomography

Human T-lymphotropic virus 1 (HTLV-1) is transmitted directly between cells via an organized cell-cell contact called a virological synapse (VS) [1], [2]. The VS has been studied by light microscopy, but the ultrastructure of the VS and the nature of the transmitted viral particle have remained unknown. Cell-free enveloped virions of HTLV-1 are undetectable in the serum of individuals infected with the human T-lymphotropic virus 1 (HTLV-1) and during in vitro culture of naturally infected lymphocytes. However, the viral envelope protein is required for infectivity of HTLV-1, suggesting that complete, enveloped HTLV-1 virions are transferred across the synapse. Here, we use electron tomography combined with immunostaining of viral protein to demonstrate the presence of enveloped HTLV-1 particles within the VS formed between naturally infected lymphocytes. We show in 3D that HTLV-1 particles can be detected in multiple synaptic clefts at different locations simultaneously within the same VS. The synaptic clefts are surrounded by the tightly apposed plasma membranes of the two cells. HTLV-1 virions can contact the recipient cell membrane before detaching from the infected cell. The results show that the HTLV-1 virological synapse that forms spontaneously between lymphocytes of HTLV-1 infected individuals allows direct cell-cell transmission of the virus by triggered, directional release of enveloped HTLV-1 particles into confined intercellular spaces