484 research outputs found
A Positivity-Preserving Finite Element Scheme for the Relaxed Cahn-Hilliard Equation with Single-Well Potential and Degenerate Mobility
We propose and analyse a finite element approximation of the Cahn-Hilliard equation regularised in space with single-well potential of Lennard-Jones type and degenerate mobility. The Cahn-Hilliard model has recently been applied to model evolution and growth for living tissues: although the choices of degenerate mobility and singular potential are biologically relevant, they induce difficulties regarding the design of a numerical scheme. We propose a finite element scheme in one and two dimensions and we show that it preserves the physical bounds of the solutions thanks to an upwind approach adapted to the finite elements method. Moreover, we show well-posedness, energy stability properties and convergence of solutions to the numerical scheme. Finally, numerical simulations in one and two dimensions are presented
A nonnegativity preserving scheme for the relaxed Cahn-Hilliard equation with single-well potential and degenerate mobility
We propose and analyze a finite element approximation of the relaxed
Cahn-Hilliard equation with singular single-well potential of Lennard-Jones
type and degenerate mobility that is energy stable and nonnegativity
preserving. The Cahn-Hilliard model has recently been applied to model
evolution and growth for living tissues: although the choices of degenerate
mobility and singular potential are biologically relevant, they induce
difficulties regarding the design of a numerical scheme. We propose a finite
element scheme and we show that it preserves the physical bounds of the
solutions thanks to an upwind approach adapted to the finite element method.
Moreover, we show well-posedness, energy stability properties, and convergence
of solutions of the numerical scheme. Finally, we validate our scheme by
presenting numerical simulations in one and two dimensions
Strict separation and numerical approximation for a non-local Cahn-Hilliard equation with single-well potential
In this paper we study a non-local Cahn-Hilliard equation with singular
single-well potential and degenerate mobility. This results as a particular
case of a more general model derived for a binary, saturated, closed and
incompressible mixture, composed by a tumor phase and a healthy phase, evolving
in a bounded domain. The general system couples a Darcy-type evolution for the
average velocity field with a convective reaction-diffusion type evolution for
the nutrient concentration and a non-local convective Cahn-Hilliard equation
for the tumor phase. The main mathematical difficulties are related to the
proof of the separation property for the tumor phase in the Cahn-Hilliard
equation: up to our knowledge, such problem is indeed open in the literature.
For this reason, in the present contribution we restrict the analytical study
to the Cahn-Hilliard equation only. For the non-local Cahn- Hilliard equation
with singular single-well potential and degenerate mobility, we study the
existence and uniqueness of weak solutions for spatial dimensions .
After showing existence, we prove the strict separation property in three
spatial dimensions, implying the same property also for lower spatial
dimensions, which opens the way to the proof of uniqueness of solutions.
Finally, we propose a well posed and gradient stable continuous finite element
approximation of the model for , which preserves the physical
properties of the continuos solution and which is computationally efficient,
and we show simulation results in two spatial dimensions which prove the
consistency of the proposed scheme and which describe the phase ordering
dynamics associated to the system
Sharp Interface Limits of the Cahn-Hilliard Equation with Degenerate Mobility
In this work, the sharp interface limit of the degenerate Cahn-Hilliard
equation (in two space dimensions) with a polynomial double well free energy
and a quadratic mobility is derived via a matched asymptotic analysis involving
exponentially large and small terms and multiple inner layers. In contrast to
some results found in the literature, our analysis reveals that the interface
motion is driven by a combination of surface diffusion flux proportional to the
surface Laplacian of the interface curvature and an additional contribution
from nonlinear, porous-medium type bulk diffusion, For higher degenerate
mobilities, bulk diffusion is subdominant. The sharp interface models are
corroborated by comparing relaxation rates of perturbations to a radially
symmetric stationary state with those obtained by the phase field model.Comment: 27 pages, 2 figure
High order finite element calculations for the deterministic Cahn-Hilliard equation
In this work, we propose a numerical method based on high degree continuous
nodal elements for the Cahn-Hilliard evolution. The use of the p-version of the
finite element method proves to be very efficient and favorably compares with
other existing strategies (C^1 elements, adaptive mesh refinement, multigrid
resolution, etc). Beyond the classical benchmarks, a numerical study has been
carried out to investigate the influence of a polynomial approximation of the
logarithmic free energy and the bifurcations near the first eigenvalue of the
Laplace operator
A second order in time, uniquely solvable, unconditionally stable numerical scheme for Cahn-Hilliard-Navier-Stokes equation
We propose a novel second order in time numerical scheme for
Cahn-Hilliard-Navier- Stokes phase field model with matched density. The scheme
is based on second order convex-splitting for the Cahn-Hilliard equation and
pressure-projection for the Navier-Stokes equation. We show that the scheme is
mass-conservative, satisfies a modified energy law and is therefore
unconditionally stable. Moreover, we prove that the scheme is uncondition- ally
uniquely solvable at each time step by exploring the monotonicity associated
with the scheme. Thanks to the weak coupling of the scheme, we design an
efficient Picard iteration procedure to further decouple the computation of
Cahn-Hilliard equation and Navier-Stokes equation. We implement the scheme by
the mixed finite element method. Ample numerical experiments are performed to
validate the accuracy and efficiency of the numerical scheme
Positivity-preserving, energy stable numerical schemes for the Cahn-Hilliard equation with logarithmic potential
We present and analyze finite difference numerical schemes for the Allen
Cahn/Cahn-Hilliard equation with a logarithmic Flory Huggins energy potential.
Both the first order and second order accurate temporal algorithms are
considered. In the first order scheme, we treat the nonlinear logarithmic terms
and the surface diffusion term implicitly, and update the linear expansive term
and the mobility explicitly. We provide a theoretical justification that, this
numerical algorithm has a unique solution such that the positivity is always
preserved for the logarithmic arguments. In particular, our analysis reveals a
subtle fact: the singular nature of the logarithmic term around the values of
and 1 prevents the numerical solution reaching these singular values, so
that the numerical scheme is always well-defined as long as the numerical
solution stays similarly bounded at the previous time step. Furthermore, an
unconditional energy stability of the numerical scheme is derived, without any
restriction for the time step size. The unique solvability and the
positivity-preserving property for the second order scheme are proved using
similar ideas, in which the singular nature of the logarithmic term plays an
essential role. For both the first and second order accurate schemes, we are
able to derive an optimal rate convergence analysis, which gives the full order
error estimate. The case with a non-constant mobility is analyzed as well. We
also describe a practical and efficient multigrid solver for the proposed
numerical schemes, and present some numerical results, which demonstrate the
robustness of the numerical schemes
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