24 research outputs found
A Robust Solver for a Second Order Mixed Finite Element Method for the Cahn-Hilliard Equation
We develop a robust solver for a second order mixed finite element splitting
scheme for the Cahn-Hilliard equation. This work is an extension of our
previous work in which we developed a robust solver for a first order mixed
finite element splitting scheme for the Cahn-Hilliard equaion. The key
ingredient of the solver is a preconditioned minimal residual algorithm (with a
multigrid preconditioner) whose performance is independent of the spacial mesh
size and the time step size for a given interfacial width parameter. The
dependence on the interfacial width parameter is also mild.Comment: 17 pages, 3 figures, 4 tables. arXiv admin note: substantial text
overlap with arXiv:1709.0400
On the nonlocal Cahn-Hilliard-Brinkman and Cahn-Hilliard-Hele-Shaw systems
The phase separation of an isothermal incompressible binary fluid in a porous
medium can be described by the so-called Brinkman equation coupled with a
convective Cahn-Hilliard (CH) equation. The former governs the average fluid
velocity , while the latter rules evolution of , the
difference of the (relative) concentrations of the two phases. The two
equations are known as the Cahn-Hilliard-Brinkman (CHB) system. In particular,
the Brinkman equation is a Stokes-like equation with a forcing term (Korteweg
force) which is proportional to , where is the chemical
potential. When the viscosity vanishes, then the system becomes the
Cahn-Hilliard-Hele-Shaw (CHHS) system. Both systems have been studied from the
theoretical and the numerical viewpoints. However, theoretical results on the
CHHS system are still rather incomplete. For instance, uniqueness of weak
solutions is unknown even in 2D. Here we replace the usual CH equation with its
physically more relevant nonlocal version. This choice allows us to prove more
about the corresponding nonlocal CHHS system. More precisely, we first study
well-posedness for the CHB system, endowed with no-slip and no-flux boundary
conditions. Then, existence of a weak solution to the CHHS system is obtained
as a limit of solutions to the CHB system. Stronger assumptions on the initial
datum allow us to prove uniqueness for the CHHS system. Further regularity
properties are obtained by assuming additional, though reasonable, assumptions
on the interaction kernel. By exploiting these properties, we provide an
estimate for the difference between the solution to the CHB system and the one
to the CHHS system with respect to viscosity
A robust solver for a second order mixed finite element method for the Cahn–Hilliard equation
We develop a robust solver for a second order mixed finite element splitting scheme for the Cahn–Hilliard equation. This work is an extension of our previous work in which we developed a robust solver for a first order mixed finite element splitting scheme for the Cahn–Hilliard equation. The key ingredient of the solver is a preconditioned minimal residual algorithm (with a multigrid preconditioner) whose performance is independent of the spatial mesh size and the time step size for a given interfacial width parameter. The dependence on the interfacial width parameter is also mild
On the Cahn-Hilliard-Brinkman system
We consider a diffuse interface model for phase separation of an isothermal
incompressible binary fluid in a Brinkman porous medium. The coupled system
consists of a convective Cahn-Hilliard equation for the phase field ,
i.e., the difference of the (relative) concentrations of the two phases,
coupled with a modified Darcy equation proposed by H.C. Brinkman in 1947 for
the fluid velocity . This equation incorporates a diffuse interface
surface force proportional to , where is the so-called
chemical potential. We analyze the well-posedness of the resulting
Cahn-Hilliard-Brinkman (CHB) system for . Then we establish
the existence of a global attractor and the convergence of a given (weak)
solution to a single equilibrium via {\L}ojasiewicz-Simon inequality.
Furthermore, we study the behavior of the solutions as the viscosity goes to
zero, that is, when the CHB system approaches the Cahn-Hilliard-Hele-Shaw
(CHHS) system. We first prove the existence of a weak solution to the CHHS
system as limit of CHB solutions. Then, in dimension two, we estimate the
difference of the solutions to CHB and CHHS systems in terms of the viscosity
constant appearing in CHB
Convergence Analysis and Error Estimates for a Second Order Accurate Finite Element Method for the Cahn-Hilliard-Navier-Stokes System
In this paper, we present a novel second order in time mixed finite element
scheme for the Cahn-Hilliard-Navier-Stokes equations with matched densities.
The scheme combines a standard second order Crank-Nicholson method for the
Navier-Stokes equations and a modification to the Crank-Nicholson method for
the Cahn-Hilliard equation. In particular, a second order Adams-Bashforth
extrapolation and a trapezoidal rule are included to help preserve the energy
stability natural to the Cahn-Hilliard equation. We show that our scheme is
unconditionally energy stable with respect to a modification of the continuous
free energy of the PDE system. Specifically, the discrete phase variable is
shown to be bounded in and the discrete
chemical potential bounded in , for any time
and space step sizes, in two and three dimensions, and for any finite final
time . We subsequently prove that these variables along with the fluid
velocity converge with optimal rates in the appropriate energy norms in both
two and three dimensions.Comment: 33 pages. arXiv admin note: text overlap with arXiv:1411.524
Existence and uniqueness of global weak solutions to a Cahn-Hilliard-Stokes-Darcy system for two phase incompressible flows in karstic geometry
We study the well-posedness of a coupled Cahn-Hilliard-Stokes-Darcy system
which is a diffuse-interface model for essentially immiscible two phase
incompressible flows with matched density in a karstic geometry. Existence of
finite energy weak solution that is global in time is established in both 2D
and 3D. Weak-strong uniqueness property of the weak solutions is provided as
well