We construct a Galerkin finite element method for the numerical approximation of weak
solutions to a general class of coupled FENE-type finitely extensible nonlinear elastic
dumbbell 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
⊂ ℝ, = 2 or 3, for
the velocity...

We construct a Galerkin finite element method for the numerical approximation of weak solutions to a coupled microscopic-macroscopic bead-spring model that arises from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The model consists of the unsteady incompressible Navier–Stokes equations in a bounded domain Ω ⊂ ${\mathbb{R}}^{d}$,= 2 or 3, for the velocity and the pressure of the fluid, with an elastic extra-stress tensor as right-hand side in the momentum equation....

We consider a fully practical finite element approximation of the following degenerate system
$$\phantom{\rule{-56.9055pt}{0ex}}\frac{\partial}{\partial t}\rho \left(u\right)-\nabla .\left(\phantom{\rule{0.166667em}{0ex}}\alpha \left(u\right)\phantom{\rule{0.166667em}{0ex}}\nabla u\right)\ni \sigma \left(u\right)\phantom{\rule{0.166667em}{0ex}}{\left|\nabla \phi \right|}^{2},\phantom{\rule{1.0em}{0ex}}\nabla .\left(\phantom{\rule{0.166667em}{0ex}}\sigma \left(u\right)\phantom{\rule{0.166667em}{0ex}}\nabla \phi \right)=0$$
subject to an initial condition on the temperature, $u$, and boundary conditions on both $u$ and the electric potential, $\phi $. In the above $\rho \left(u\right)$ is the enthalpy incorporating the latent heat of melting, $\alpha \left(u\right)\>0$ is the temperature dependent heat conductivity, and $\sigma \left(u\right)\ge 0$ is the electrical conductivity. The latter is zero in the frozen zone, $u\le 0$, which gives rise to the degeneracy in this Stefan system....

Existence of a solution to the quasi-variational inequality problem arising in a model for sand surface evolution has been an open problem for a long time. Another long-standing open problem concerns determining the dual variable, the flux of sand pouring down the evolving sand surface, which is also of practical interest in a variety of applications of this model. Previously, these problems were solved for the special case in which the inequality is simply variational. Here, we introduce a regularized...

Using the approach in [5] for analysing
time discretization error and assuming
more regularity on the initial data, we improve on
the error bound derived in [2]
for a fully practical piecewise linear
finite element approximation with a backward Euler time
discretization
of a model for phase separation of a multi-component alloy with
non-smooth free energy.

We consider a fully practical finite element approximation of the
following degenerate system
$$\frac{\partial}{\partial t}\rho \left(u\right)-\nabla .\left(\phantom{\rule{0.166667em}{0ex}}\alpha \left(u\right)\phantom{\rule{0.166667em}{0ex}}\nabla u\right)\ni \sigma \left(u\right)\phantom{\rule{0.166667em}{0ex}}{\left|\nabla \phi \right|}^{2},\phantom{\rule{1.0em}{0ex}}\nabla .\left(\phantom{\rule{0.166667em}{0ex}}\sigma \left(u\right)\phantom{\rule{0.166667em}{0ex}}\nabla \phi \right)=0$$
subject to an initial condition on the temperature, ,
and boundary conditions on both
and the electric potential, .
In the above
is the enthalpy
incorporating the latent heat of melting, is
the temperature dependent heat conductivity, and
is the electrical
conductivity. The latter is zero in the frozen zone, ≤ 0,
which gives rise to the degeneracy in this Stefan system.
In addition to showing...

We consider a system
of degenerate parabolic equations modelling a
thin film, consisting of two layers of immiscible Newtonian liquids, on
a solid horizontal substrate.
In addition, the model includes the presence of insoluble surfactants on
both the free liquid-liquid and liquid-air interfaces,
and the presence of both attractive and repulsive van der Waals forces
in terms of the heights of the two layers.
We show that this system formally satisfies a Lyapunov structure,
and a second energy...

We construct a Galerkin finite element method for the numerical approximation of weak solutions to a coupled microscopic-macroscopic bead-spring model that arises from the kinetic theory of dilute solutions
of polymeric liquids with noninteracting polymer chains. The model consists of the unsteady incompressible Navier–Stokes equations in a bounded domain Ω ⊂ ${\mathbb{R}}^{d}$, = 2 or 3, for the velocity and
the pressure of the fluid, with an elastic extra-stress tensor as right-hand side in the momentum equation....

We construct a Galerkin finite element method for the numerical approximation of weak
solutions to a general class of coupled FENE-type finitely extensible nonlinear elastic
dumbbell 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
⊂ ℝ, = 2 or 3, for
the velocity...

We introduce and analyse a mixed formulation of the
Monge-Kantorovich equations, which express optimality conditions for
the mass transportation problem with cost proportional to distance.
Furthermore, we introduce and analyse the finite element
approximation of this formulation using the lowest order
Raviart-Thomas element. Finally, we present some numerical
experiments, where both the optimal transport density and the
associated Kantorovich potential are computed for a coupling problem
and problems...

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