• Abstract

In this paper, we develop an adaptive high-order surface finite element method (FEM) incorporating spectral deferred correction method for chain contour discretization to solve polymeric self-consistent field equations on general curved surfaces. The high-order surface FEM is obtained by the high-order surface geometrical approximation and the high-order function space approximation. Numerical results demonstrate that the precision order of these methods is consistent with theoretical prediction. In order to describe the sharp interface in the strongly segregated system more accurately, an adaptive FEM equipped with a new \textit{Log} marking strategy is proposed. The \textit{Log} marking strategy can not only label the elements that need to be refined or coarsened, but also give the refined or coarsened times, which can make full use of the information of a posterior error estimator and improve the efficiency of the adaptive algorithm. To demonstrate the power of our approach, we investigate the self-assembled patterns of diblock copolymers on several distinct curved surfaces.  Numerical results illustrate the efficiency of the proposed method, especially for strongly segregated systems with economical discretization nodes.

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In this paper, we propose a nonconvex and nonsmooth model for image restoration with Cauchy noise removal. The regularization term is composed of a nonconvex Lipschitz potential function with first-order derivatives of an image, while the data fidelity term is introduced based on the maximum a posteriori estimator to Cauchy distribution. To handle the nonconvexity of regularizers, we adopt the proximal linearization technique to convert the original nonconvex model to a series of convex models, which can be easily implemented by alternating direction method with multipliers. Based on the Kurdyka-{\L}ojasiewicz property, we can verify the global convergence of the proposed algorithm by using an abstract convergence framework.

Numerical experiments and comparisons indicate that our method obtains good restorations and is effective for better preserving edges.

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Consider a Urysohn integral equation $x - \mathcal{K} (x) = f$, where $f$ and the integral operator $\mathcal{K}$ with kernel of the type of Green's function are given. In the computation of approximate solutions of the given integral equation by Galerkin method, all the integrals are needed to be evaluated by some numerical integration formula. This gives rise to the discrete version of the Galerkin method. For $r \geq 1$, a space of piecewise polynomials of degree $\leq r-1$ with respect to a uniform partition is chosen to be the approximating space. For the appropriate choice of a numerical integration formula, an asymptotic series expansion of the discrete iterated Galerkin solution is obtained at the above partition points. Richardson extrapolation is used to improve the order of convergence. Using this method we can restore the rate of convergence when the error is measured in the continuous case. Numerical examples are given to illustrate this theory.

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This paper is devoted to adaptive finite element method for the nonlinear Schr\"{o}dinger equation. The adaptive method is based on the extrapolation technology and a second order accurate, linear and mass preserving finite element scheme. For error control, we take the difference between the numerical gradient and the recovered gradient obtained by the superconvergent cluster recovery method as the spatial discretization error estimator and the difference of numerical approximations between two consecutive time steps as the temporal discretization error estimator. A time-space adaptive algorithm is developed for numerical approximation of the nonlinear Schr\"{o}dinger equation. Numerical experiments are presented to illustrate the reliability and efficiency of the proposed error estimators and the corresponding adaptive algorithm.

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In this paper, an exponential transformation based lattice Boltzmann (LB) model for solving the $n$-dimensional ($n$D) convection-diffusion equation(CDE) is developed. Firstly, a class of exponential transformation is proposed to convert the $n$D CDE into a diffusion equation. Then, the converted diffusion equation is solved by the LB model. So, compared to the available LB models for CDE, the present LB model can eliminate the difficulty in treating the convection term. With the direct Taylor expansion method, it is shown that the CDE can be exactly derived from the exponential transformation based LB model. Finally, a variety of numerical tests have been conducted to validate the present LB model. It can be found that the numerical results agree well with the analytical solutions. Moreover, we also find that the present LB model has second-order convergence rate in space, and it is more effective and more stable than the previous LB model for the CDE.

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Model Order Reduction is an approximation of the main system so that the simplified system retains important features of the main system. Deep learning technology is a recent breakthrough in artificial neural networks that can find more hidden information from the data. In this paper, a non-intrusive reduced order model (NIROM) based on combining deep neural networks (DNNs) and POD abilities, namely FAE-CAE-LSTM is presented. This method combines the obtained features based on Fully connected autoencoders (FAE), Convolutional autoencoders (CAE), and POD and then, a deep Long short-term memory network is trained by obtained features to predict the pressure and velocity fields at future time instances. We investigate the performance of the proposed methodology by solving two well-known canonical cases: a strong shear flow exhibiting the Kelvin–Helmholtz instability, and flow past a cylinder. The performance of the proposed FAE-CAE-LSTM method in future state prediction of the flow is compared with other NIROM methods such as CAE-LSTM, autoencoder-LSTM, autoencoder-DMD and POD-RNN based models. Results show that the FAE-CAE-LSTM method is considerably capable of predicting fluid flow evolution and obtains the best results in the prediction of the pressure and velocity fields in future time instances.

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The incompressible magnetohydrodynamics system with variable density is coupled by the incompressible Navier-Stokes equations with variable density and the Maxwell equations. In this paper, we study a new first-order Euler semi-discrete scheme for solving this system. The proposed numerical scheme is unconditionally stable for any time step size $\tau>0$. Furthermore, a rigorous error analysis is presented and the first-order temporal convergence rate $\mathcal O(\tau)$ is derived by using the method of mathematical induction and the discrete maximal $L^p$-regularity of the Stokes problem. Finally, numerical results are given to support the theoretical analysis.

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The stochastic Allen-Cahn equations, as a typical example of nonlinear stochastic partial differential equations, play an important role in phase theory. In this paper, we investigate the rate of convergence in the $p$th mean for a truncated-type explicit Euler time-stepping method applied to the stochastic Allen-Cahn equations as well as using the spectral Galerkin approximation in spatial discretization. Finally, a numerical example is given to confirm the strong convergence order.

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A well-conditioned spectral integration (SI) method is introduced, developed and applied to $n$th-order differential equations with variable coefficients and general boundary conditions. The approach is based on integral reformulation techniques which lead to almost banded linear matrices, and the main system to be solved is further banded by utilizing a Schur complement approach. Numerical experiments indicate the spectral integration method can solve high order equations efficiently, oscillatory problems accurately and is adaptable to large systems. Applications in Korteweg-de Vries (KdV) type and Kawahara equations are carried out to illustrate the proposed method is effective to complicated mathematical models.

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In this article, we have developed a new time-marching method to simulate sound and soliton propagation in homogeneous and heterogeneous media. The proposed time integration scheme can numerically preserve physical dispersion over a wide wavenumber range and conserve energy while solving wave propagation problems. Here, the Fourier stability analysis

has been used to assess the numerical properties of the developed method. The proposed numerical method's dispersion and dissipation properties have also been compared with the classical fourth-order Runge-Kutta (RK4) method. Stability property contours for the newly proposed method display that the maximum allowable time step is at least five times higher than the RK4 method. The Fourier stability analysis also explains the dispersion error associated with the used spatio-temporal discretization schemes. It is observed that the dispersion error is significantly small for the proposed time integration schemes compared with the RK4 method. The proposed

methods simulate sound propagation problems with fewer computational resources that otherwise demand high computational costs. The efficacy of the proposed time integration methods has been demonstrated by solving

benchmark sound wave propagation problems. Moreover, to test the developed method's efficiency and robustness, we have performed simulations of the sound wave propagation in a layered media, corner-edge model, and damped sine-Gordon equation.

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In this paper, the first-order and second-order semi-discrete predictor-multicorrector (PMC) algorithms to solve the 2D/3D unsteady incompressible micropolar fluid equations (IMNSE) are proposed.  In the algorithms, the first-order and second-order BDF formulas are adopted to approximate the time derivative terms. At each time step,  two elliptical sub-problems with Dirichlet conditions are solved at the prediction step, the strategy of projection about linear momentum equation with additional viscosity term and the elliptical sub-problems about angular momentum are solved at the multicorrection step. Furthermore, the unconditional stability and error estimates of the first-order scheme  are proved theoretically. Numerical experiments are carried out to show the effectiveness of the algorithms.

• General Summary