We consider a C0 interior penalty finite element approximation of a sixth-order Cahn-Hilliard type equation that models the dynamics of phase transitions in ternary oil-water-surfactant systems. The nonlinear sixth-order parabolic equation is expressed in a mixed form whereby a second-order (in space) parabolic equation and an algebraic fourth-order (in space) nonlinear equation are considered. The temporal discretization is chosen so that a discrete energy law can be established leading to unconditional energy stability. Additionally, we show that the numerical method is unconditionally uniquely solvable. We conclude with several numerical experiments demonstrating the unconditional stability and first-order accuracy of the proposed method.

In this paper, we consider a two-dimensional linear nonlocal model involving a singular matrix kernel. For the initial value problem, we first give well-posedness results and energy conservation via Fourier transform. Meanwhile, we also discuss the corresponding Dirichlet-type nonlocal boundary value problems in the cases of both positive and semi-positive definite kernels, where the core is the coercivity of bilinear forms. In addition, in the limit of vanishing nonlocality, the solution of the nonlocal model is seen to converge to a solution of its classical elasticity local model provided that $c_t = 0.$

In this work, we study theoretically and numerically the non-stationary Navier-Stokes’s equations under power law slip boundary condition. We establish existence of a unique solution by using a semi-discretization in time combined with the weak convergence approach. Next, we formulate and analyze the discretization in time and the finite element approximation in space associated to the continuous problem. We derive optimal convergence in time and space provided that the solution is regular enough on the slip zone. Iterative schemes for solving the nonlinear problems is formulated and convergence is studied. Numerical experiments presented confirm the theoretical findings.

Nesterov’s accelerated forward-backward algorithm (AFBA) is an efficient algorithm for solving a class of two-term convex optimization models consisting of a differentiable function with a Lipschitz continuous gradient plus a nondifferentiable function with a closed form of its proximity operator. It has been shown that the iterative sequence generated by AFBA with a modified Nesterov’s momentum scheme converges to a minimizer of the objective function with an $o (\frac{1}{k^2})$ convergence rate in terms of the function value (FV-convergence rate) and an $o(\frac{1}{k})$ convergence rate in terms of the distance between consecutive iterates (DCI-convergence rate). In this paper, we propose a more general momentum scheme with an introduced power parameter $ω ∈ (0, 1]$ and show that AFBA with the proposed momentum scheme converges to a minimizer of the objective function with an $o ( \frac{1}{ k^{2ω}} )$ FV-convergence rate and an $o (\frac{1}{k^ω})$ DCI-convergence rate. The generality of the proposed momentum scheme provides us a variety of parameter selections for different scenarios, which makes the resulting algorithm more flexible to achieve better performance. We then employ AFBA with the proposed momentum scheme to solve the smoothed hinge loss $ℓ_1$-support vector machine model. Numerical results demonstrate that the proposed generalized momentum scheme outperforms two existing momentum schemes.

In this article, the implicit-explicit (IMEX) compact schemes are proposed to solve the partial integro-differential equations (PIDEs), and the linear complementarity problems (LCPs) arising in option pricing. A diagonally dominant tri-diagonal system of linear equations is achieved for a fully discrete problem by eliminating the second derivative approximation using the variable itself and its first derivative approximation. The stability of the fully discrete problem is proved using Schur polynomial approach. Moreover, the problem’s initial condition is smoothed to ensure the fourth-order convergence of the proposed IMEX compact schemes. Numerical illustrations for solving the PIDEs and the LCPs with constant and variable coefficients are presented. For each case, obtained results are compared with the IMEX finite difference scheme, and it is observed that proposed approach significantly outperforms the finite difference scheme.

In this article, we give a penalty finite element method for the steady-state closed-loop geothermal model. Firstly, we construct the stationary penalty closed-loop geothermal equations. Secondly, we propose a finite element method for the penalty system and deduce error estimates. Finally, some numerical experiments are used to illustrate the theoretical results of the presented method.

We develop a splitting spectral method for the time-dependent nonlinear Dirac-Poisson (DP) equations. Through time splitting method, we split the time-dependent nonlinear DP equations into linear and nonlinear subproblems. To advance DP from time $t_n$ to $t_{n+1},$ the nonlinear subproblem can be integrated analytically, and linear Dirac and Poisson equation are well resolved by Fourier and Sine spectral method respectively. Compared with conventional numerical methods, our method achieves spectral accuracy in space, conserves total charge on the discrete level. Extensive numerical results confirm the spatial spectral accuracy, the second order temporal accuracy, and the $l^2$-stable property. Finally, an application from laser field is proposed to simulate the spin-flip phenomenon.