816 research outputs found
Time-fractional Cahn-Hilliard equation: Well-posedness, degeneracy, and numerical solutions
In this paper, we derive the time-fractional Cahn-Hilliard equation from
continuum mixture theory with a modification of Fick's law of diffusion. This
model describes the process of phase separation with nonlocal memory effects.
We analyze the existence, uniqueness, and regularity of weak solutions of the
time-fractional Cahn-Hilliard equation. In this regard, we consider
degenerating mobility functions and free energies of Landau, Flory--Huggins and
double-obstacle type. We apply the Faedo-Galerkin method to the system, derive
energy estimates, and use compactness theorems to pass to the limit in the
discrete form. In order to compensate for the missing chain rule of fractional
derivatives, we prove a fractional chain inequality for semiconvex functions.
The work concludes with numerical simulations and a sensitivity analysis
showing the influence of the fractional power. Here, we consider a convolution
quadrature scheme for the time-fractional component, and use a mixed finite
element method for the space discretization
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
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Analysis of a finite element formulation for modelling phase separation
In Combescure, A., De Borst, R., and Belytschko, T., editors, IUTAM Symposium on Discretization Methods for Evolving Discontinuities, volume 5 of IUTAM Bookseries, pages 89–102. Springer.The Cahn-Hilliard equation is of importance in materials science and a range of other fields. It represents a diffuse interface model for simulating the evolution of phase separation in solids and fluids, and is a nonlinear fourth-order parabolic equation, which makes its numerical solution particularly challenging. To this end, a finite element formulation has been developed which can solve the Cahn-Hilliard equation in its primal form using C^0 basis functions. Here, analysis of a fully discrete version of this method is presented in the form of a priori uniqueness, stability and error analysis
A free-energy stable nodal discontinuous Galerkin approximation with summation-by-parts property for the Cahn-Hilliard equation
We present a nodal Discontinuous Galerkin (DG) scheme for the Cahn-Hilliard
equation that satisfies the summation-by-parts simultaneous-approximation-term
(SBP-SAT) property. The latter permits us to show that the discrete free-energy
is bounded, and as a result, the scheme is provably stable. The scheme and the
stability proof are presented for general curvilinear three-dimensional
hexahedral meshes. We use the Bassi-Rebay 1 (BR1) scheme to compute interface
fluxes, and an IMplicit-EXplicit (IMEX) scheme to integrate in time. Lastly, we
test the theoretical findings numerically and present examples for two and
three-dimensional problems
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