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research-article

Analysis and Approximation of a Fractional Cahn--Hilliard Equation

Authors:

Mark Ainsworth,

Zhiping MaoAuthors Info & Claims

SIAM Journal on Numerical Analysis, Volume 55, Issue 4

Pages 1689 - 1718

https://doi.org/10.1137/16M1075302

Published: 01 January 2017 Publication History

Abstract

We derive a fractional Cahn--Hilliard equation (FCHE) by considering a gradient flow in the negative order Sobolev space $H^{-\alpha}$, $\alpha\in [0,1]$, where the choice $\alpha=1$ corresponds to the classical Cahn--Hilliard equation while the choice $\alpha=0$ recovers the Allen--Cahn equation. The existence of a unique solution is established and it is shown that the equation preserves mass for all positive values of fractional order $\alpha$ and that it indeed reduces the free energy. We then turn to the delicate question of the $L_\infty$ boundedness of the solution and establish an $L_\infty$ bound for the FCHE in the case where the nonlinearity is a quartic polynomial. As a consequence of the estimates, we are able to show that the Fourier--Galerkin method delivers a spectral rate of convergence for the FCHE in the case of a semidiscrete approximation scheme. Finally, we present results obtained using computational simulation of the FCHE for a variety of choices of fractional order $\alpha$. It is observed that the nature of the solution of the FCHE with a general $\alpha>0$ is qualitatively (and quantitatively) closer to the behavior of the classical Cahn--Hilliard equation than to the Allen--Cahn equation, regardless of how close to zero the value of $\alpha$ is. An examination of the coarsening rates of the FCHE reveals that the asymptotic rate is rather insensitive to the value of $\alpha$ and, as a consequence, is close to the well-established rate observed for the classical Cahn--Hilliard equation.

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Index Terms

Analysis and Approximation of a Fractional Cahn--Hilliard Equation
1. Mathematics of computing
  1. Mathematical analysis
2. Theory of computation
  1. Design and analysis of algorithms
    1. Approximation algorithms analysis

Index terms have been assigned to the content through auto-classification.

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cover image SIAM Journal on Numerical Analysis

SIAM Journal on Numerical Analysis Volume 55, Issue 4

DOI:10.1137/sjnaam.55.4

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© 2017, Society for Industrial and Applied Mathematics.

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Society for Industrial and Applied Mathematics

United States

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Published: 01 January 2017

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Cited By

Jiang DAzaiez MMiranville AXu CYao H(2024)On the long-time behavior of the continuous and discrete solutions of a nonlocal Cahn–Hilliard type inpainting modelMathematics and Computers in Simulation10.1016/j.matcom.2024.05.023225:C(461-479)Online publication date: 18-Oct-2024
https://dl.acm.org/doi/10.1016/j.matcom.2024.05.023
Yang ZChen XChen YWang J(2024)Accurate numerical simulations for fractional diffusion equations using spectral deferred correction methodsComputers & Mathematics with Applications10.1016/j.camwa.2023.11.001153:C(123-129)Online publication date: 1-Jan-2024
https://dl.acm.org/doi/10.1016/j.camwa.2023.11.001
Zhang PJiang XJia J(2024)Improved uniform error estimates for the two-dimensional nonlinear space fractional Dirac equation with small potentials over long-time dynamicsApplied Mathematics and Computation10.1016/j.amc.2023.128458466:COnline publication date: 1-Apr-2024
https://dl.acm.org/doi/10.1016/j.amc.2023.128458
Jia JJiang XChi X(2024)Improved uniform error bounds of Lawson-type exponential integrator method for long-time dynamics of the high-dimensional space fractional sine-Gordon equationNumerical Algorithms10.1007/s11075-023-01745-097:3(1179-1214)Online publication date: 1-Nov-2024
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Hu DFu YCai WWang Y(2023)Unconditional Convergence of Conservative Spectral Galerkin Methods for the Coupled Fractional Nonlinear Klein–Gordon–Schrödinger EquationsJournal of Scientific Computing10.1007/s10915-023-02108-694:3Online publication date: 8-Feb-2023
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Huang XLi DSun HZhang F(2022)Preconditioners with Symmetrized Techniques for Space Fractional Cahn-Hilliard EquationsJournal of Scientific Computing10.1007/s10915-022-01900-092:2Online publication date: 1-Aug-2022
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