In this paper, both the finite element method (FEM) and the mesh-free deep neural network (DNN) approach are studied in a comparative fashion for solving two types of coupled nonlinear hyperbolic/wave partial differential equations (PDEs) in a space of high dimension $\mathbb{R}^d (d > 1),$ where the first PDE system to be studied is the coupled nonlinear Korteweg-De Vries (KdV) equations modeling the solitary wave and waves on shallow water surfaces, and the second PDE system is the coupled nonlinear Klein-Gordon (KG) equations modeling solitons as well as solitary waves. A fully connected, feedforward, multi-layer, mesh-free DNN approach is developed for both coupled nonlinear PDEs by reformulating each PDE model as a least-squares (LS) problem based upon DNN-approximated solutions and then optimizing the LS problem using a $(d + 1)$-dimensional space-time sample point (training) set. Mathematically, both coupled nonlinear hyperbolic problems own significant differences in their respective PDE theories; numerically, they are approximated by virtue of a fully connected, feedforward DNN structure in a uniform fashion. As a contrast, a distinct and sophisticated FEM is developed for each coupled nonlinear hyperbolic system, respectively, by means of the Galerkin approximation in space and the finite difference scheme in time to account for different characteristics of each hyperbolic PDE system. Overall, comparing with the subtly developed, problem-dependent FEM, the proposed mesh-free DNN method can be uniformly developed for both coupled nonlinear hyperbolic systems with ease and without a need of mesh generation, though, the FEM can produce a concrete convergence order with respect to the mesh size and the time step size, and can even preserve the total energy for KG equations, whereas the DNN approach cannot show a definite convergence pattern in terms of parameters of the adopted DNN structure but only a universal approximation property indicated by a relatively small error that rarely changes in magnitude, let alone the dissipation of DNN-approximated energy for KG equations. Both approaches have their respective pros and cons, which are also validated in numerical experiments by comparing convergent accuracies of the developed FEMs and approximation performances of the proposed mesh-free DNN method for both hyperbolic/wave equations based upon different types of discretization parameters changing in doubling, and specifically, comparing discrete energies obtained from both approaches for KG equations.

The numerical analysis of the coupled Cahn-Hilliard/Allen-Cahn system endowed with dynamic boundary conditions is studied in this article. We consider a semi-discretisation in space using a finite element method and we derive error estimates between the exact and the approximate solution. Then, using the backward Euler scheme for the time variable, a fully discrete scheme is obtained and its stability is proved. Some numerical simulations illustrate the behavior of the solution under the influence of dynamical boundary conditions.

A new embedded low-regularity integrator is proposed for the quadratic nonlinear Schrödinger equation on the one-dimensional torus. Second-order convergence in $H^\gamma$ is proved for solutions in $C([0, T]; H^\gamma)$ with $\gamma > \frac{3}{2},$ i.e., no additional regularity in the solution is required. The proposed method is fully explicit and can be computed by the fast Fourier transform with $\mathcal{(O} log N)$ operations at every time level, where $N$ denotes the degrees of freedom in the spatial discretization. The method extends the first-order convergent low-regularity integrator in [14] to second-order time discretization in the case $\gamma >\frac{3}{2}$ without requiring additional regularity of the solution. Numerical experiments are presented to support the theoretical analysis by illustrating the convergence of the proposed method.

In this paper, we are interested in the analysis of the convergence of a low order mixed finite element method for the Monge-Ampère equation. The unknowns in the formulation are the scalar variable and the discrete Hessian. The distinguished feature of the method is that the unknowns are discretized using only piecewise linear functions. A superconvergent gradient recovery technique is first applied to the scalar variable, then a piecewise gradient is taken, the projection of which gives the discrete Hessian matrix. For the analysis we make a discrete elliptic regularity assumption, supported by numerical experiments, for the discretization based on gradient recovery of an equation in non divergence form. A numerical example which confirms the theoretical results is presented.

This paper studies the virtual element method for Stokes problem with a least-squares type stabilization. The method cannot only circumvent the Babuška-Brezzi condition, but also make use of general polygonal meshes, as opposed to more standard triangular grids. Moreover, it is suitable for arbitrary combinations of the velocity and pressure, including equal-order virtual element. We obtain the corresponding energy norm error estimates and $L^2$ norm error estimates for velocity. Finally, a series of numerical experiments are performed to verify the method has good behaviors.

As an important pre-processing step for many related computer vision tasks, color image denoising has attracted considerable attention in image processing. However, traditional methods often regard the red, green, and blue channels of color images independently without considering the correlations among the three channels. In order to overcome this deficiency, this paper proposes a novel dictionary method for color image denoising based on pure quaternion representation, which efficiently deals with both single-channel and cross-channel information. The pure quaternion constraint is firstly used to force the sparse representations of color images to contain only red, green, and blue color information. Moreover, a total variation regularization is proposed in the quaternion domain and embedded into the pure quaternion-based representation model, which is effective to recover the sharp edges of color images. To solve the proposed model, a new numerical scheme is also developed based on the alternating minimization method (AMM). Experimental results demonstrate that the proposed model has better denoising results than the state-of-the-art methods, including a deep learning approach DnCNN, in terms of PSNR, SSIM, and visual quality.