65Zxx Applications to physics
An adaptive a posteriori error estimator based finite element method for the
numerical solution of a coupled Cahn-Hilliard/Navier-Stokes system with a
double-obstacle homogenous free (interfacial) energy density is proposed. A
semi-implicit Euler scheme for the time-integration is applied which results in
a system coupling a quasi-Stokes or Oseen-type problem for the fluid flow to
a variational inequality for the concentration and the chemical potential according
to the Cahn-Hilliard model [13]. A Moreau-Yosida regularization is
employed which relaxes the constraints contained in the variational inequality
and, thus, enables semi-smooth Newton solvers with locally superlinear convergence
in function space. Moreover, upon discretization this yields a mesh
independent method for a fixed relaxation parameter. For the finite dimensional
approximation of the concentration and the chemical potential piecewise
linear and globally continuous finite elements are used, and for the numerical
approximation of the fluid velocity Taylor-Hood finite elements are employed.
The paper ends by a report on numerical examples showing the efficiency of the
new method
One of the most challenging problems in dynamic concurrent multiscale
simulations is the reflectionless transfer of physical quantities between the
different scales. In particular, when coupling molecular dynamics and finite
element discretizations in solid body mechanics, often spurious wave reflections
are introduced by the applied coupling technique. The reflected waves are
typically of high frequency and are arguably of little importance in the domain
where the finite element discretization drives the simulation.
In this work, we provide an analysis of this phenomenon.
Based on the gained
insight, we derive a new coupling approach, which neatly separates high and low
frequency waves. Whereas low frequency waves are permitted to
bridge the scales, high frequency waves can be removed by applying damping techniques without affecting the coupled share of the solution. As a consequence, our new method almost completely eliminates unphysical wave reflections and deals in a consistent way with waves of arbitrary frequencies. The separation of
wavelengths is achieved by employing a discrete $L^2$-projection, which acts as a
low pass filter. Our coupling constraints enforce matching in the range of this projection. With respect to the numerical realization this approach
has the advantage of a small number of constraints, which is computationally
efficient. Numerical results in one and two dimensions confirm our theoretical
findings and illustrate the performance of our new weak coupling approach.