scholarly journals Efficient Fully Discrete Finite-Element Numerical Scheme with Second-Order Temporal Accuracy for the Phase-Field Crystal Model

Mathematics ◽  
2022 ◽  
Vol 10 (1) ◽  
pp. 155
Author(s):  
Jun Zhang ◽  
Xiaofeng Yang
Keyword(s):  
Finite Element ◽  
Phase Field ◽  
Second Order ◽  
The Finite Element Method ◽  
Fully Discrete ◽  
Crystal Model ◽  
Time Marching ◽  
The Stability ◽  
Discrete Finite Element ◽  
Phase Field Crystal

In this paper, we consider numerical approximations of the Cahn–Hilliard type phase-field crystal model and construct a fully discrete finite element scheme for it. The scheme is the combination of the finite element method for spatial discretization and an invariant energy quadratization method for time marching. It is not only linear and second-order time-accurate, but also unconditionally energy-stable. We prove the unconditional energy stability rigorously and further carry out various numerical examples to demonstrate the stability and the accuracy of the developed scheme numerically.

A MFE method combined with L1-approximation for a nonlinear time-fractional coupled diffusion system

2017 ◽  
Vol 08 (01) ◽  
pp. 1750012 ◽  
Author(s):  
Yaxin Hou ◽  
Ruihan Feng ◽  
Yang Liu ◽  
Hong Li ◽  
Wei Gao
Keyword(s):  
Finite Element ◽  
Mixed Finite Element ◽  
A Priori ◽  
Diffusion System ◽  
Fully Discrete ◽  
Coupled Diffusion ◽  
L1 Approximation ◽  
Priori Error Estimates ◽  
The Stability ◽  
Discrete Finite Element

In this paper, a nonlinear time-fractional coupled diffusion system is solved by using a mixed finite element (MFE) method in space combined with L1-approximation and implicit second-order backward difference scheme in time. The stability for nonlinear fully discrete finite element scheme is analyzed and a priori error estimates are derived. Finally, some numerical tests are shown to verify our theoretical analysis.

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.

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