Error estimates for time discretizations of Cahn–Hilliard and Allen–Cahn phase-field models for two-phase incompressible flows

Cai, Yongyong; Choi, Heejun; Shen, Jie

doi:10.1007/s00211-017-0875-9

Cited by 37 publications

(7 citation statements)

References 23 publications

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“…The Cahn-Hilliard has the added difficulty of a fourth order derivative in space. Although there exists an extensive literature on numerical methods for these models in planar and volumetric domains (see, e.g., recent publications [46,80,61,19,48] and references therein), there are not so many papers where the equations are treated on surfaces. Solving equations numerically on general surfaces poses additional difficulties that are related to the discretization of tangential differential operators and the approximate recovery of complex shapes.…”

Section: Introductionmentioning

confidence: 99%

A computational study of lateral phase separation in biological membranes

Yushutin

Quaini

Majd³

et al. 2019

Numer Methods Biomed Eng

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Conservative and non-conservative phase-field models are considered for the numerical simulation of lateral phase separation and coarsening in biological membranes. An unfitted finite element method is devised for these models to allow for a flexible treatment of complex shapes in the absence of an explicit surface parametrization. For a set of biologically relevant shapes and parameter values, the paper compares the dynamic coarsening produced by conservative and non-conservative numerical models, its dependence on certain geometric characteristics and convergence to the final equilibrium.

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Section: Introductionmentioning

confidence: 99%

A computational study of lateral phase separation in biological membranes

Yushutin

Quaini

Majd³

et al. 2019

Numer Methods Biomed Eng

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“…The Cahn-Hilliard equation is challenging to solve numerically due to non-linearity, stiffness, and the presence of a fourth order derivative in space. For some recent publications on the CH equation in planar and volumetric domains, we refer to [28,64,38,10] and references therein. The numerical solution of the CH equation posed on surfaces is further complicated by the need to discretize tangential differential operators and to approximately recover complex shapes.…”

mentioning

confidence: 99%

Numerical modeling of phase separation on dynamic surfaces

Yushutin

Quaini

Olshanskii

2020

Journal of Computational Physics

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The paper presents a model of lateral phase separation in a two component material surface. The resulting fourth order nonlinear PDE can be seen as a Cahn-Hilliard equation posed on a time-dependent surface. Only elementary tangential calculus and the embedding of the surface in R 3 are used to formulate the model, thereby facilitating the development of a fully Eulerian discretization method to solve the problem numerically. A hybrid method, finite difference in time and trace finite element in space, is introduced and stability of its semi-discrete version is proved. The method avoids any triangulation of the surface and uses a surface-independent background mesh to discretize the equation. Thus, the method is capable of solving the Cahn-Hilliard equation numerically on implicitly defined surfaces and surfaces undergoing strong deformations and topological transitions. We assess the approach on a set of test problems and apply it to model spinodal decomposition and pattern formation on colliding surfaces. Finally, we consider the phase separation on a sphere splitting into two droplets.

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“…Since ε is often very small, there is also a vast literature on constructing energy-decaying time-stepping methods for phase field models with large time stepsizes (compared with ε), including stabilized semi-implicit schemes [15,16,35,42], invariant energy quadratization methods [46][47][48], and the scalar auxiliary variable approach [41] and [3].…”

Section: Georgios Akrivis and Buyang LImentioning

confidence: 99%

Error estimates for fully discrete BDF finite element approximations of the Allen–Cahn equation

Akrivis

2020

IMA Journal of Numerical Analysis

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For a class of compatible profiles of initial data describing bulk phase regions separated by transition zones, we approximate the Cauchy problem of the Allen–Cahn (AC) phase field equation by an initial-boundary value problem in a bounded domain with the Dirichlet boundary condition. The initial-boundary value problem is discretized in time by the backward difference formulae (BDF) of order $1\leqslant q\leqslant 5$ and in space by the Galerkin finite element method of polynomial degree $r-1$, with $r\geqslant 2$. We establish an error estimate of $O(\tau ^q\varepsilon ^{-q-\frac 12}+h^{r}\varepsilon ^{-r-\frac 12}+{e}^{-c/\varepsilon })$ with explicit dependence on the small parameter $\varepsilon$ describing the thickness of the phase transition layer. The analysis utilizes the maximum-norm stability of BDF and finite element methods with respect to the boundary data, the discrete maximal $L^p$-regularity of BDF methods for parabolic equations and the Nevanlinna–Odeh multiplier technique combined with a time-dependent inner product motivated by a spectrum estimate of the linearized AC operator.

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Error estimates for time discretizations of Cahn–Hilliard and Allen–Cahn phase-field models for two-phase incompressible flows

Cited by 37 publications

References 23 publications

A computational study of lateral phase separation in biological membranes

A computational study of lateral phase separation in biological membranes

Numerical modeling of phase separation on dynamic surfaces

Error estimates for fully discrete BDF finite element approximations of the Allen–Cahn equation

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