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

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.

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

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 Triangular Spectral Element Method for the 2-D Viscous Burgers Equation

A triangular spectral element method is established for the two-dimensional viscous Burgers equation. In the spatial direction, a new type of mapping is applied. We splice the local basis functions on each triangle into a global basis function. The second-order Crank-Nicolson/ leap-frog (CNLF) method is used for discretization in the time direction. Due to the use of a quasi-interpolation operator, the nonlinear term can be handled conveniently. We give the fully discrete scheme of the method and the implementation process of the algorithm. Numerical examples verify the effectiveness of this method.

Numerical Analysis of Least-Squares Group Finite Element Method for Coupled Burgers' Problem

In this paper, a least squares group finite element method for solving coupled Burgers' problem in 2-D is presented. A fully discrete formulation of least squares finite element method is analyzed, the backward-Euler scheme for the time variable is considered, the discretization with respect to space variable is applied as biquadratic quadrangular elements with nine nodes for each element. The continuity, ellipticity, stability condition and error estimate of least squares group finite element method are proved. The theoretical results show that the error estimate of this method is . The numerical results are compared with the exact solution and other available literature when the convection-dominated case to illustrate the efficiency of the proposed method that are solved through implementation in MATLAB R2018a.