Here we consider the numerical approximations of the 2D simplified Ericksen-Leslie system. We first rewrite the system and get a new system. For the new system, we propose an easy-to-implement time discretization scheme which preserves the sphere constraint at each node, enjoys a discrete energy law, and leads to linear and decoupled elliptic equations to be solved at each time step. A discrete maximum principle of the scheme in the finite element form is also proved. Some numerical simulations are performed to validate the scheme and simulate the dynamic motion of liquid crystals.

This paper designs a hybrid scheme based on finite difference methods and a spectral method for the time-dependent Wigner equation, and gives the error analysis for the full discretization of its initial value problem. An explicit-implicit time-splitting scheme is used for time integration and the second-order upwind finite difference scheme is used to discretize the advection term. The consistence error and the stability of the full discretization are analyzed. A Fourier spectral method is used to approximate the pseudo-differential operator term and the corresponding error is studied in detail. The final convergence result shows clearly how the regularity of the solution affects the convergence order of the proposed scheme. Numerical results are presented for confirming the sharpness of the analysis. The scattering effects of a Gaussian wave packet tunneling through a Gaussian potential barrier are investigated. The evolution of the density function shows that a larger portion of the wave is reflected when the height and the width of the barrier increase.

Efficient and accurate Chebyshev dual-Petrov-Galerkin methods for solving first-order equation, third-order equation, third-order KdV equation and fifth-order Kawahara equation are proposed. Some Sobolev bi-orthogonal basis functions are constructed which lead to the diagonalization of discrete systems. Accordingly, both the exact solutions and the approximate solutions are expanded as an infinite and truncated Fourier-like series, respectively. Numerical experiments illustrate the effectiveness of the suggested approaches.

In this paper, the superconvergence properties of the time-dependent Navier-Stokes equations are investigated by a low order nonconforming mixed finite element method (MFEM). In terms of the integral identity technique, the superclose error estimates for both the velocity in broken $H^1$-norm and the pressure in $L^2$-norm are first obtained, which play a key role to bound the numerical solution in $L^{\infty}$-norm. Then the corresponding global superconvergence results are derived through a suitable interpolation postprocessing approach. Finally, some numerical results are provided to demonstrate the theoretical analysis.

This article introduces a novel variational model for restoring images degraded by Cauchy noise and/or blurring. The model integrates a nonconvex data-fidelity term with two regularization terms, a sparse representation prior over dictionary learning and total generalized variation (TGV) regularization. The sparse representation prior exploiting patch information enables the preservation of fine features and textural patterns, while adequately denoising in homogeneous regions and contributing natural visual quality. TGV regularization further assists in effectively denoising in smooth regions while retaining edges. By adopting the penalty method and an alternating minimization approach, we present an efficient iterative algorithm to solve the proposed model. Numerical results establish the superiority of the proposed model over other existing models in regard to visual quality and certain image quality assessments.

This paper deals with the numerical computation and analysis for Caputo fractional differential equations (CFDEs). By combining the $p$-order boundary value methods (BVMs) and the $m$-th Lagrange interpolation, a type of extended BVMs for the CFDEs with $\gamma$-order ($0 <\gamma < 1$) Caputo derivatives are derived. The local stability, unique solvability and convergence of the methods are studied. It is proved under the suitable conditions that the convergence order of the numerical solutions can arrive at $\min\left\{p, m-\gamma+1\right\}$. In the end, by performing several numerical examples, the computational efficiency, accuracy and comparability of the methods are further illustrated.

This paper gives the detailed numerical analysis of mixed finite element method for fractional Navier-Stokes equations. The proposed method is based on the mixed finite element method in space and a finite difference scheme in time. The stability analyses of semi-discretization scheme and fully discrete scheme are discussed in detail. Furthermore, We give the convergence analysis for both semidiscrete and fully discrete schemes and then prove that the numerical solution converges the exact one with order $O(h^2+k)$, where $h$ and $k$ respectively denote the space step size and the time step size. Finally, numerical examples are presented to demonstrate the effectiveness of our numerical methods.

We study a class of deep neural networks with architectures that form a directed acyclic graph (DAG). For backpropagation defined by gradient descent with adaptive momentum, we show weights converge for a large class of nonlinear activation functions. The proof generalizes the results of Wu et al. (2008) who showed convergence for a feed-forward network with one hidden layer. For an example of the effectiveness of DAG architectures, we describe an example of compression through an AutoEncoder, and compare against sequential feed-forward networks under several metrics.