We design and analyze an iterative two-grid algorithm for the finite element discretizations of strongly nonlinear elliptic boundary value problems in this paper. We propose an iterative two-grid algorithm, in which a nonlinear problem is first solved on the coarse space, and then a symmetric positive definite problem is solved on the fine space. The main contribution in this paper is to establish a first convergence analysis, which requires dealing with four coupled error estimates, for the iterative two-grid methods. We also present some numerical experiments to confirm the efficiency of the proposed algorithm.

In this paper, an effective oscillation-free discontinuous Galerkin (DG) scheme for a nonlinear stochastic convection-dominated problem is formulated and analyzed. The proposed oscillation-free scheme is capable to capture the steep fronts of solution automatically and distinguish the influence of the convection domination and noise perturbation. Under proper regularity assumptions, the optimal convergence rates in space and time are rigorously proved with the techniques of variational solution and conditional expectation. In the numerical simulation, the classical SIPG scheme and the proposed oscillation-free DG scheme are both performed and compared. The numerical convergence rates tests are first carried out to verify the theoretical results. The benchmark tests having the steep behaviors are further provided to illustrate the effectiveness and robustness of our proposed oscillation-free DG scheme.

This paper is devoted to the study of a sub-diffusion equation involving a Clarke subdifferential boundary condition. It describes transport of particles governed by the anomalous diffusion in media with boundary semipermeability. The weak formulation of the model problem results in a time fractional parabolic hemivariational inequality. We first construct an abstract hemivariational evolutionary inclusion and prove its unique solvability using a time-discretization approximation, known as the Rothe method. In addition, a numerical approach based on a finite difference scheme in time and finite dimensional approximation in space is proposed and analyzed for the abstract problem. These results are then applied to establish the convergence of the numerical solution of the model problem. Under appropriate regularity assumptions, an optimal order error estimate for the linear finite element method is derived. Some numerical examples are provided to support the theoretical results.

This paper investigates an interface-coupled fractional subdiffusion model, featuring two subdiffusion equations in adjacent domains connected by an interface allowing bidirectional energy transfer. The fractional derivative, accounting for long-term medium effects, introduces challenges in theoretical analysis and computational efficiency. We propose a partitioned time-stepping algorithm using higher-order extrapolations on the interface term to decouple the system with improved temporal accuracy, combined with finite element spatial approximations. Rigorous theoretical analysis demonstrates unconditional stability and optimal $L^2$ norm error estimates, supported by several numerical experiments.

In this paper, the novel optimization model for solving tensor completion with noise is proposed, its objective function is a convex combination of the minimum nuclear norm and maximum nuclear norm. The necessary condition and sufficient condition of the stationary point and optimal solution are discussed. Based on the proximal gradient algorithm and feasible direction method, we design the new algorithm for solving the proposed nonconvex and nonsmooth optimization problem and prove that the sub-sequence generated by the new algorithm converges to the stationary point. Finally, experimental results on the random sample completions and images show that the proposed optimization and algorithm are superior to the compared algorithms in CPU time or precision.

In this paper, we propose an adaptive algorithm for L1-fidelity color image restoration by using saturation-value total variation. The main contribution of this paper is to employ the generalized cross validation method efficiently and automatically to estimate the regularization parameter in a saturation-value total variation plus L1-fidelity color image restoration model. We consider Poisson noise and mixed noise in this paper, and the experimental results show that the visual quality and the SSIM/PSNR/SAM values of the restored images by using the proposed algorithm are competitive with other tested existing methods, which makes the proposed algorithm to be comparable both quantitatively and qualitatively.

In this research article, we present convenient analytical-approximate solutions for fluid flow models known as multi-dimensional Navier-Stokes equations containing time-fractional order by using a relatively new analytical method called modified generalized Mittag-Leffler function method. The Caputo fractional derivative is used to describe fractional mathematical formalism. The approximate solutions for five problems are implemented to demonstrate the validity and accuracy of the proposed method. It is also demonstrated that the solutions obtained from our method when $α = 1$ coincide with the exact solutions, this is displayed by using some 2D and 3D plots for each problem. Moreover, the comparison between our outcomes with given exact solutions and results obtained by other methods in the literature besides absolute error is provided in some tables. Additionally, we offer some plots when $α$ has different values to present the effect of fractional order on the solution of each suggested problem. The numerical simulation presented in this work indicates that the proposed method is efficient, reliable, accurate and easy which has less computational ability to give analytical-approximate solution form. So, this method can be extended to implement on different related problems arising in various areas of innovation and research.

In this paper, our focus is on examining the robustness of the central scheme in two dimensions. Although stability analyses are available in the literature for the scheme’s solution of scalar conservation laws, the associated Courant-Friedrichs-Lewy (CFL) number is often notably small, occasionally degenerating to zero. This challenge is traced back to the initial data reconstruction. The interface value limiter used in the reconstruction proves insufficient to maintain the invariant region of the updated solutions. To overcome this limitation, we introduce the vertex value limiter, resulting in a more suitable CFL number that is half of the one-dimensional value. We present a unified analysis of stability applicable to both types of limiters. This enhanced stability condition enables the utilization of larger time steps, offering improved resolution to the solution and ensuring faster simulations. Our analysis extends to general conservation laws, encompassing scalar problems and nonlinear systems. We support our findings with numerical examples, validating our claims and showcasing the robustness of the enhanced scheme.

In this paper, we introduce a nonlinear finite volume scheme preserving discrete strong extremum principle (DSEP) for diffusion equations on tetrahedral meshes. In the construction of our nonlinear scheme, the key is to reformulate a discrete normal flux with local extremum principle structure, which is based on a modification of a second order linear scheme. In the construction of existing cell-centered finite volume schemes that maintain the discrete maximum principle, it is required to represent auxiliary unknowns as convex combinations of primary unknowns, which results in strong constraints on the smoothness of the mesh and diffusion coefficient. By contrast, our new scheme avoids this kind of constraints. Moreover, we will prove that there holds the DSEP for any solution of our scheme and there exists at least one solution preserving DSEP for our scheme. Furthermore, a modified Picard iteration with the Anderson acceleration (mP-AA) for solving the nonlinear scheme is proposed, and the nonlinear convergence of the modified Picard iteration is also proved. Finally, numerical examples are presented to show that the new scheme preserves DSEP and obtains second order accuracy, as well as the mP-AA method is effective.