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In this paper, we present the design of a posteriori error estimator for the plasmon phenomenon on the graphene surface and propose a method to achieve local high-precision numerical calculations when plasmon phenomena occur on the graphene surface. We provide a lower bound estimate for the posteriori error estimator, along with a proof of convergence. Specifically, the constructed posterior error estimator enables local refinement in regions where the error is significant at the graphene interface. Firstly, we outline the construction of the posterior error estimator and provide the proof of its lower bound. Secondly, we establish the convergence of the Adaptive Edge Finite Element Method  (AEFEM). Finally, we present numerical results that validate the effectiveness of the error estimator.

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In this paper we study the nonconforming Crouzeix-Raviart element and the enriched Crouzeix-Raviart element methods firstly for the transmission eigenvalue problem of inhomogeneous media and the modified transmission eigenvalue problem in inverse scattering. Using $\mathbb{T}$-coercivity method, we prove the convergence and the a priori error estimate of approximate eigenpair for the transmission eigenvalue problem, and based on the obtained results we prove the a priori error estimate for the modified transmission eigenvalue problem  by the $\mathbb{T}$-coercivity method and G{\aa}rding inequality, and further prove that the discrete eigenvalues for the problem with metamaterial background approximate the exact eigenvalue from above. We also carry out numerical experiments to validate the theoretical findings and the efficiency of the proposed methods.

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Dynamic stall poses a significant aerodynamic challenge for vertical axis wind turbines (VAWTs) and serves as a key impediment to enhancing their overall efficiency. Consequently, to elucidate the underlying mechanism of dynamic stall and facilitate a clearer understanding of the entire dynamic stall process, a numerical investigation is conducted over the blades of a VAWT system. A two-dimensional numerical simulation is conducted by employing unsteady Reynolds-averaged Navier-Stokes (RANS) calculations with a Reynolds stress turbulence model. Four real-time key indicators are generalized to quantitatively characterize dynamic stall over the VAWTs. These indicators have been demonstrated to accurately predict the initiation and detachment times of the dynamic stall vortex (DSV), providing deeper insights into the underlying mechanism of DSV development. Additionally, this study explores the impact of blade thickness on the dynamic stall of VAWTs. Three airfoils from the NACA 4-digit series, each featuring distinct ratios of thickness to chord length, are examined. As the blade cross-section transitions from NACA 0012 to NACA 0015 and NACA 0018, heightened blade thickness results in a delayed onset of dynamic stall, shifting from leading edge stall to trailing edge stall. Moreover, the blade thickness significantly influences the efficiency of VAWT system, with increased airfoil thickness contributing to an improved utilization of wind energy.

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In this paper, we propose a new inexact iterative subspace projection method for real symmetric positive definite generalized eigenvalue problems based on the WYD (an abbreviation of the initials of the proposers: Wilson, Yuan, and Dickens [1]) method. Firstly an analysis of the convergence condition of the approximate eigenvectors is given when ${\bf Ar}_{k+1}=\lambda{\bf Br}_k$ is inexactly solved by the Krylov subspace methods. Then the inexact iterative WYD (IIWYD) method is constructed, which utilizes the approximate Ritz subspace generated by the WYD method with an inexact solver as the main search space. The IIWYD method improves the quality of the search space during the iterative process, significantly reduces the number of iteration steps, and improves the overall computational efficiency and stability. The results of numerical experiments show that the IIWYD method is more efficient and stable compared to the locally optimal block preconditioned conjugate gradient (LOBPCG) method and the Jacobi-Davidson (JD) method. In addition, we also discuss the effects of the refined strategy and the conjugate strategy in our method.

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Based on the generalized scalar auxiliary variable approach and vector penalty projection method, some fully discrete schemes with first- and second-order accuracy in time direction are constructed for solving the incompressible magnetohydrodynamic model. It is a combination of mixed finite element approximation for spatial discretization and first-order backward Euler/second-order backward differential formula for temporal discretization. The proposed schemes own several features: it decouples unknown physical variables and linearizes the nonlinear terms, then it only needs to solve some linear equations at each temporal level; although the divergence of numerical velocity is not exactly equal to zero, it can approximately meet the mass conservation when one takes small penalty parameter; while the computation of the velocity and pressure are decoupled, numerical results show that the velocity and pressure can reach second-order accuracy in time. The resulting schemes are supported by numerical analysis and simulation.

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In this paper, we develop through a careful selection of the auxiliary variables and numerical fluxes an energy conservative local discontinuous Galerkin (LDG) method based on a hybrid form of the general Euler-Korteweg (EK) equations with a variable capillarity coefficient. This energy conservative LDG discretization is of optimal order of accuracy for alternating numerical fluxes, but not for central numerical fluxes which result in the reduction of one order of accuracy when odd degree polynomial basis functions are used. Also, a relatively simple energy conservative LDG discretization for the EK-equations with an irrotational velocity field is presented. Due to the presence of a highly nonlinear third-order spatial derivative term, which originates from the divergence of the Korteweg stress tensor, we employ the novel semi-implicit spectral deferred correction (SDC) method as temporal discretization. The SDC method can be applied to highly nonlinear ordinary differential equations (ODEs) without separating stiff and non-stiff components and is numerically stable for a time step proportional to the mesh size. Numerical experiments, including ones with adaptive meshes, are performed to illustrate the accuracy and capability of the proposed methods to solve the EK-equations.

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A new extrapolation cascadic multigrid (EXCMG) method is developed to solve large sparse symmetric positive definite systems resulting from the classical 25-point finite-difference discretizations of the three-dimensional (3D) biharmonic equation on rectangular domains. We accomplish this by designing a quartic interpolation-based prolongation operator and using the symmetric successive over-relaxation (SSOR) preconditioned CG method as the multigrid smoother. For the new prolongation operator, quartic interpolations are used for the finite difference solutions on coarse and fine grids twice and once so that two approximations can be obtained on the next finer grid,  and then the completed Richardson extrapolation is used for these two approximations to obtain an excellent initial guess on the next finer grid. The proposed EXCMG method with the new prolongation operator is easier to implement than the original EXCMG method. Numerical experiments demonstrate that the new EXCMG is a highly efficient solver for the 3D biharmonic equation and is considerably faster than the original EXCMG method and the aggregation-based algebraic multigrid (AGMG) method developed by Y. Notay. The proposed EXCMG method can solve discrete 3D biharmonic equations with more than 100 million unknowns in dozens of seconds.

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The mean field games (MFG) theory has broad application in mathematical modeling of social phenomena. The Mean Field Games System (MFGS) is the key to the MFG theory. This is a system of two nonlinear parabolic partial differential equations with two opposite directions of time $t\in (0,T). $ The topic of Coefficient Inverse Problem (CIPs) for the MFGS is a newly emerging one. A CIP for the MFGS is studied. The input data are Dirichlet and Neumann boundary conditions either on a part of the lateral boundary (incomplete data) or on the whole lateral boundary (complete data). In addition to the initial conditions at $\left\{t=0\right\}, $ terminal conditions at $\left\{t=T\right\} $ are given. The terminal conditions mean the final overdetermination. The necessity of assigning all these input data is explained. H\"{o}lder and Lipschitz stability estimates are obtained for the cases of incomplete and complete data respectively. These estimates imply uniqueness of the CIP.

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In this paper, a pointwise goal-oriented residual-based a posteriori error estimator is proposed for linear elliptic equations with restricted source terms. The pointwise error is directly estimated by introducing the dual problem with a Dirac delta source term instead of using classical mollification technique. The goal-oriented error estimator is proved to be the upper bound of the pointwise error. Numerical experiments show the advantage of the adaptive finite element method (AFEM) based on this error estimator, which can preserve the monotonicity of the pointwise error, compared with the goal-oriented AFEM using the mollification technique.

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This paper proposes a direct Eulerian generalized Riemann problem (GRP) scheme for two-layer shallow water equations.  The model takes into account the distinctions between different densities and velocities, and is obtained by taking the vertical averaging across the layer depth. The source terms generated from the mass and momentum exchange prevent us from solving the Riemann problem analytically. We consider an equivalent conservative two-layer model which describes the horizontal velocity with two degrees of freedom. The rarefaction wave and the shock wave are analytically resolved by using the Riemann invariants and Rankine-Hugoniot condition, respectively. Numerical simulations are also given on some typical problems in order to verify the good performance of the GRP method.

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Recently, Jin et al. proposed a quantum simulation technique for  any  linear partial differential equations (PDEs), called  Schrodingerisation [1-3].  In this paper,   the Schrodingerisation technique for quantum simulation is expanded to  any nonlinear PDEs by combining it with the homotopy analysis method (HAM) [4-6].  The HAM  can transfer a nonlinear PDE into a series of linear PDEs with guaranteeing  convergence of the series.   In this way,  any nonlinear PDEs  can be solved by quantum simulation  using a quantum computer.  For simplicity, we call the procedure "HAM-Schrodingerisation quantum algorithm''.   Quantum computing is a groundbreaking technique.  Hopefully, the  "HAM-Schrodingerisation quantum algorithm'' can open a door to highly efficient simulation  of complicated turbulent flows  by  means  of  quantum  computing  in future.

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Image smoothing techniques are widely used in computer vision and graphics applications. In this paper, we present an $\ell_p$-nonconvex regularization method for image smoothing. To induce sparsity prior of the smoothed images more strongly than the $\ell_1$ norm regularization, we take the nonconvex arctangent penalty function of the image gradient as the regularization term. To make the proposed model more flexible and effective for different image smoothing applications, we use the $\ell_p$ norm function as the fidelity term, instead of the $\ell_2$ norm function. The powerful majorization-minimization (MM) algorithm is employed for the proposed nonconvex optimization model. The convergence of the resulting MM algorithm is discussed. Comprehensive experiments and comparisons show that the proposed method is effective in various image processing tasks such as texture smoothing, detail enhancement, artifact removal, image denoising, high dynamic range (HDR) tone mapping, edge detection, and image composition.

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In this paper, we develop one-level and multilevel meshfree radial basis functions (RBF) collocation methods for solving the Helmholtz equation with variable wave speed on a bounded connected Lipschitz domain. The approximate solution is constructed by employing successive refinement scattered data sets and scaled compactly supported radial basis functions with varying support radii. We prove the convergence of one-level and multilevel collocation method for the modelling problem in Sobolev spaces. The convergence rates depend on the regularity of the solution, the smoothness of the computational domain, the bounds of frequency and wave speed, the approximation of scaled kernel-based spaces, the increasing rules of scattered data, and the selection of scaling parameters. 

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In this work, by introducing a new family of recursively defined processes, we propose new explicit multistep schemes for coupled second-order forward backward stochastic differential equations. The explicit schemes avoid calculating the conditional mathematical expectations of the generator $f$ and calculate the required values of $f$ explicitly and accurately. By combining the Sinc quadrature rule for approximating the conditional expectations, we further propose the $k$th order ($1\le k\le 6$) fully discrete explicit multistep schemes. Numerical tests are presented to demonstrate the strong stability, high accuracy, and high efficiency of the explicit schemes.

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In this paper, we propose a weak Galerkin finite element method (WG) for solving singularly perturbed convection-diffusion problems on a Bakhvalov-type mesh in 2D. Our method is flexible and allows the use of discontinuous approximation functions on the mesh. An error estimate is developed in a suitable norm, and the optimal convergence order is obtained. Finally, numerical experiments are conducted to support the theory and to demonstrate the efficiency of the proposed method.

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In this paper, we study the asymptotic properties  of  the singularly perturbed subdiffusion equations in a bounded domain. First, we use the matched asymptotic expansion method to obtain the uniform asymptotic expansion for the solution $u(x,t)$ of the singularly perturbed subdiffusion equation. By asymptotic analysis, we can know that near the boundary configured with non-smooth boundary values, the solution $u(x,t)$ of the singularly perturbed subdiffusion equation has a boundary layer of thickness $\mathcal{O}(\varepsilon)$. By studying the asymptotic properties of the spatial partial derivatives $\partial_xu(x,t)$ and $\partial_{xx}u(x,t)$, we can know that the singularity is mainly concentrated in the boundary layers, and then  the solution $u(x,t)$ changes gently outside the boundary layers. Next, we introduce a new $\mathcal{L}1$-TFPM scheme to solve the singularly perturbed subdiffusion equations numerically.  Some numerical experiments can demonstrate the correctness of the asymptotic analysis results.

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In this paper, a robust and globally divergence-free weak Galerkin finite element method of Oseen equations is proposed and analyzed. We use the $\mathbf{P}_{k}/P_{k-1}$ discontinuous finite element combination for the approximation of velocity and pressure, and piecewise $\mathbf{P}_{k}/P_{k}$ for the numerical traces of velocity and pressure. This method not only yields globally divergence-free velocity approximations, but is also robust in the sense that a priori error estimates are uniform with respect to the coefficients of Oseen equations, providing the exact solutions are sufficiently smooth. Finally, numerical examples are given to confirm our theoretical results.

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We consider the nonconforming discrete Raviart-Thomas mixed finite element method (dRT-MFEM) for obstacle problems with $p$-Laplacian differential operator. The a posteriori and a priori error analysis were presented in a new sense of measurement. A number of experiments confirm the effective decay rates of the proposed dRT-MFEM.

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In this study, a class of explicit structure-preserving Du Fort-Frankel-type FDMs are firstly developed for Fisher-Kolmogorov-Petrovsky-Piscounov (Fisher-KPP) equation. They inherit some properties of the continuous problems, such as

non-negativity, maximum principle and monotonicity. Besides, by using the discrete maximum principle, the error estimate in $L^{\infty}$-norm is proven to be $\mathcal{O}(\tau+h_{x}^{2}+h_{y}^{2}+(\frac{\tau}{h_{x}})^{2} (\frac{\tau}{h_{y}})^{2})$ as some suitable conditions are satisfied. Here, $\tau$, $h_{x}$ and $h_{y}$ are time step and spatial meshsizes in $x$- and $y$- directions,

respectively. Then, as the current FDMs are used to solve Allen-Cahn equation, the obtained numerical solutions satisfy the discrete maximum principle and the discrete energy-dissipation law. Our methods are easy to be implemented because of explicitness. Finally, numerical results confirm theoretical findings and the efficiency of our methods.

• General Summary