A Linear Iteration Algorithm for a Second-Order Energy Stable Scheme for a Thin Film Model Without Slope Selection

Chen, Wenbin; Wang, Cheng; Wang, Xiaoming; Wise, Steven M.

doi:10.1007/s10915-013-9774-0

Cited by 92 publications

(66 citation statements)

References 23 publications

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“…Consider the following energy functional:

E [ϕ] : = \int_{normalΩ} (- \frac{1}{2} \ln false(1 + | \nabla ϕ |^{2} false) + \frac{ϵ^{2}}{2} {false(Δ ϕ false)}^{2}) d x .

The NSS dynamical equation is the L 2 gradient flow with respect to this energy:

\partial_{t} ϕ = - μ, μ : = {normalδ}_{ϕ} E = \nabla \cdot (\frac{\nabla ϕ}{1 + | \nabla ϕ |^{2}}) + {normalϵ}^{2} Δ^{2} ϕ .

Some previous works have described and analyzed second‐order accurate energy stable numerical schemes for the NSS equation (2.23). In particular, it has been demonstrated in two recent works that, as the nonlinear term has automatically

L^{\infty}

‐bounded higher order derivatives in this model, either a linear scheme or a linear iteration algorithm could be efficiently designed for the NSS equation (2.23), with energy stability theoretically justified. In other words, the logarithmic nature of the energy functional for the NSS model enables one to derive linear schemes to obtain second order temporal accuracy and energy stability, so that a complicated nonlinear solver may be avoided.…”

Section: The Fully Discrete Scheme With Finite Difference Spatial Dismentioning

confidence: 99%

A second‐order energy stable backward differentiation formula method for the epitaxial thin film equation with slope selection

Feng

Cheng²,

Wise

et al. 2018

Numerical Methods Partial

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In this article, we study a new second‐order energy stable Backward Differentiation Formula (BDF) finite difference scheme for the epitaxial thin film equation with slope selection (SS). One major challenge for higher‐order‐in‐time temporal discretizations is how to ensure an unconditional energy stability without compromising numerical efficiency or accuracy. We propose a framework for designing a second‐order numerical scheme with unconditional energy stability using the BDF method with constant coefficient stabilizing terms. Based on the unconditional energy stability property that we establish, we derive an ℓ ∞ ( 0 , T ; H h 2 ) stability for the numerical solution and provide an optimal convergence analysis. To deal with the highly nonlinear four‐Laplacian term at each time step, we apply efficient preconditioned steepest descent and preconditioned nonlinear conjugate gradient algorithms to solve the corresponding nonlinear system. Various numerical simulations are presented to demonstrate the stability and efficiency of the proposed schemes and solvers. Comparisons with other second‐order schemes are presented.

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“…Consider the following energy functional:

E [ϕ] : = \int_{normalΩ} (- \frac{1}{2} \ln false(1 + | \nabla ϕ |^{2} false) + \frac{ϵ^{2}}{2} {false(Δ ϕ false)}^{2}) d x .

The NSS dynamical equation is the L 2 gradient flow with respect to this energy:

\partial_{t} ϕ = - μ, μ : = {normalδ}_{ϕ} E = \nabla \cdot (\frac{\nabla ϕ}{1 + | \nabla ϕ |^{2}}) + {normalϵ}^{2} Δ^{2} ϕ .

L^{\infty}

Section: The Fully Discrete Scheme With Finite Difference Spatial Dismentioning

confidence: 99%

A second‐order energy stable backward differentiation formula method for the epitaxial thin film equation with slope selection

Feng

Cheng²,

Wise

et al. 2018

Numerical Methods Partial

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“…In more detail, a convex splitting numerical scheme, which treats the terms of the variational derivative implicitly or explicitly according to whether the terms corresponding to the convex or concave parts of the energy, was formulated in [19], with a mixed finite element approximation in space. Such a numerical approach assures two mathematical properties: unique solvability and unconditional energy stability; also see the related works for various PDE systems, including the phase field crystal (PFC) equation [4,5,27,34,35,39], epitaxial thin film growth model [8,10,31,33], and others [21,22]. Moreover, for a gradient system coupled with fluid motion, the idea of convex splitting can still be applied and these distinguished mathematical properties are retained, as given by a few recent works [9,12,13,19,38].…”

Section: Definition 11 Definementioning

confidence: 99%

Error analysis of a mixed finite element method for a Cahn–Hilliard–Hele–Shaw system

et al. 2016

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We present and analyze a mixed finite element numerical scheme for the Cahn-Hilliard-Hele-Shaw equation, a modified Cahn-Hilliard equation coupled with the Darcy flow law. This numerical scheme was first reported in Feng and Wise (SIAM J Numer Anal 50:1320-1343, 2012), with the weak convergence to a weak solution proven. In this article, we provide an optimal rate error analysis. A convex splitting approach is taken in the temporal discretization, which in turn leads to the unique solvability and unconditional energy stability. Instead of the more standard ∞ (0, T ; L 2 ) ∩ 2 (0, T ; H 2 ) error estimate, we perform a discrete ∞ (0, T ; H 1 ) ∩ 2 (0, T ; H 3 ) error estimate for the phase variable, through an L 2 inner product with the numerical error function associated with the chemical potential. As a result, an unconditional convergence (for the time step τ in terms of the spatial resolution h) is derived. The nonlinear analysis is accomplished with the help of a discrete Gagliardo-Nirenberg type inequality in the finite element space, gotten by introducing a discrete Laplacian h of the numerical solution, such that h φ ∈ S h , for every φ ∈ S h , where S h is the finite element space. B Steven M. Wise

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“…Setting

t_{n} = n τ

, we consider the first‐order fully discrete convex splitting finite element scheme of the Cahn–Hilliard equation: Find

(u_{h}^{n}, w_{h}^{n})

such that

(d_{t} u_{h}^{n}, q_{h}) + (\nabla w_{h}^{n}, \nabla q_{h}) = 0, \forall q_{h} \in X_{h},

(\nabla u_{h}^{n}, \nabla v_{h}) + \frac{1}{ϵ^{2}} (f (u_{h}^{n}) + u_{h}^{n} - u_{h}^{n - 1}, v_{h}) = (w_{h}^{n}, v_{h}), \forall v_{h} \in X_{h},

u_{h}^{0} (x) = R_{h} u_{0} (x),

where

d_{t} u_{h}^{n} = (u_{h}^{n} - u_{h}^{n - 1}) / τ

. For other similar convex splitting schemes applied to different physical models of gradient flow, the reader refers to .…”

Section: Energy Stability Of the First Order Fully Discrete Finite Elmentioning

confidence: 99%

“…For other similar convex splitting schemes applied to different physical models of gradient flow, the reader refers to [14,[23][24][25][26][27][28].…”

Section: Energy Stability Of the First Order Fully Discrete Finitmentioning

confidence: 99%

Error estimates of fully discrete finite element solutions for the 2D Cahn–Hilliard equation with infinite time horizon

Chen

Feng

2016

Numerical Methods Partial

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In this article, we deal with a rigorous error analysis for the finite element solutions of the two‐dimensional Cahn–Hilliard equation with infinite time. The L 2 − H 1 error estimates with respect to ( h , τ ) are proven for the fully discrete conforming piecewise linear element solution under Assumption (A1) on the initial value and Assumption (A2) on the discrete spectrum estimate in the finite element space. The analysis is based on sharp a‐priori estimates for the solutions, particularly reflecting their behavior as t → ∞ . Numerical experiments are carried out to support the theoretical analysis and demonstrate the efficiency of the fully discrete mixed finite element methods. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 742–762, 2017

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A Linear Iteration Algorithm for a Second-Order Energy Stable Scheme for a Thin Film Model Without Slope Selection

Cited by 92 publications

References 23 publications

A second‐order energy stable backward differentiation formula method for the epitaxial thin film equation with slope selection

A second‐order energy stable backward differentiation formula method for the epitaxial thin film equation with slope selection

Error analysis of a mixed finite element method for a Cahn–Hilliard–Hele–Shaw system

Error estimates of fully discrete finite element solutions for the 2D Cahn–Hilliard equation with infinite time horizon

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