scholarly journals Fully-discrete, decoupled, second-order time-accurate and energy stable finite element numerical scheme  of the Cahn-Hilliard binary surfactant model confined in the Hele-Shaw cell

2022 ◽  
Author(s):  
Xiaofeng Yang
Keyword(s):  
Finite Element ◽  
Phase Field Model ◽  
Second Order ◽  
Energy Stability ◽  
Energy Potential ◽  
Time Step ◽  
Decoupling Method ◽  
Fully Discrete ◽  
Time Accuracy ◽  
Hele Shaw Cell

We consider the numerical approximation of the binary fluid surfactant phase-field model confined in a Hele-Shaw cell, where the system includes two coupled Cahn-Hilliard equations and Darcy equations. We develop a fully-discrete finite element scheme with some desired characteristics, including linearity, second-order time accuracy, decoupling structure, and unconditional energy stability. The scheme is constructed by combining the projection method for the Darcy equation, the quadratization approach for the nonlinear energy potential, and a decoupling method of using a trivial ODE built upon the ``{zero-energy-contribution}" feature. The advantage of this scheme is that not only can all variables be calculated in a decoupled manner, but each equation has only constant coefficients at each time step. We strictly prove that the scheme satisfies the unconditional energy stability and give a detailed implementation process. Various numerical examples are further carried out to prove the effectiveness of the scheme, in which the benchmark Saffman-Taylor fingering instability problems in various flow regimes are simulated to verify the weakening effects of surfactant on surface tension.

scholarly journals Discrete Maximum Principle and Energy Stability of Compact Difference Scheme for the Allen-Cahn Equation

2018 ◽  
Author(s):  
Dan Tian ◽  
Yuanfeng Jin ◽  
Gang Lv
Keyword(s):  
Maximum Principle ◽  
Difference Scheme ◽  
Numerical Solutions ◽  
Energy Stability ◽  
Discrete Maximum Principle ◽  
Compact Difference Scheme ◽  
Time Step ◽  
Fully Discrete ◽  
Cahn Equation ◽  
Compact Difference

In the paper, a fully discrete compact difference scheme with $O(\tau^{2}+h^{4})$ precision is established by considering the numerical approximation of the one-dimensional Allen-Cahn equation. The numerical solutions satisfy discrete maximum principle under reasonable step ratio and time step constraint is proved. And the energy stability for the fully discrete scheme is investigated. An example is finally presented to show the effectiveness of scheme.

scholarly journals Longtime behavior of a second order finite element scheme simulating the kinematic effects in liquid crystal dynamics

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Mouhamadou Samsidy Goudiaby ◽  
Ababacar Diagne ◽  
Leon Matar Tine
Keyword(s):  
Finite Element ◽  
Liquid Crystal ◽  
Type Inequality ◽  
Mesh Size ◽  
Time Step ◽  
Finite Element Scheme ◽  
Element Scheme ◽  
Fully Discrete ◽  
Discrete Finite Element ◽  
Longtime Behavior

<p style='text-indent:20px;'>We consider an unconditional fully discrete finite element scheme for a nematic liquid crystal flow with different kinematic transport properties. We prove that the scheme converges towards a unique critical point of the elastic energy subject to the finite element subspace, when the number of time steps go to infinity while the time step and mesh size are fixed. A Lojasiewicz type inequality, which is the key for getting the time asymptotic convergence of the whole sequence furnished by the numerical scheme, is also derived.</p>

A numerical method for two-dimensional nonlinear modified time-fractional fourth-order diffusion equation

2019 ◽  
Vol 10 (01) ◽  
pp. 1941005 ◽  
Author(s):  
Haiyan He ◽  
Kaijie Liang ◽  
Baoli Yin
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Diffusion Equation ◽  
Fourth Order ◽  
Time Derivative ◽  
Second Order ◽  
Two Dimensional ◽  
The Finite Element Method ◽  
Discrete Scheme ◽  
Fully Discrete

In this paper, we consider the finite element method for two-dimensional nonlinear modified time-fractional fourth-order diffusion equation. In order to avoid using higher order elements, we introduce an intermediate variable [Formula: see text] and translate the fourth-order derivative of the original problem into a second-order coupled system. We discretize the fractional time derivative terms by using the [Formula: see text]-approximation and discretize the first-order time derivative term by using the second-order backward differentiation formula. In the fully discrete scheme, we implement the finite element method for the spatial approximation. Unconditional stability of the fully discrete scheme is proven and its optimal convergence order is obtained. Numerical experiments are carried out to demonstrate our theoretical analysis.

scholarly journals A New Linearized Crank-Nicolson Mixed Element Scheme for the Extended Fisher-Kolmogorov Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Jinfeng Wang ◽  
Hong Li ◽  
Siriguleng He ◽  
Wei Gao ◽  
Yang Liu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Mixed Finite Element ◽  
A Priori ◽  
Mixed Finite Element Method ◽  
Order Equation ◽  
Second Order ◽  
Fully Discrete ◽  
Priori Error Estimates ◽  
Element Method

We present a new mixed finite element method for solving the extended Fisher-Kolmogorov (EFK) equation. We first decompose the EFK equation as the two second-order equations, then deal with a second-order equation employing finite element method, and handle the other second-order equation using a new mixed finite element method. In the new mixed finite element method, the gradient∇ubelongs to the weaker(L2(Ω))2space taking the place of the classicalH(div;Ω)space. We prove some a priori bounds for the solution for semidiscrete scheme and derive a fully discrete mixed scheme based on a linearized Crank-Nicolson method. At the same time, we get the optimal a priori error estimates inL2andH1-norm for both the scalar unknownuand the diffusion termw=−Δuand a priori error estimates in(L2)2-norm for its gradientχ=∇ufor both semi-discrete and fully discrete schemes.

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