Journal of Computational and Applied Mathematics最新文献

In this paper, we consider a time-dependent discrete network model with highly varying connectivity. The approximation by time is performed using an implicit scheme. We propose the coarse scale approximation construction of network models based on the Generalized Multiscale Finite Element Method. An accurate coarse-scale approximation is generated by solving local spectral problems in sub-networks. Convergence analysis of the proposed method is presented for semi-discrete and discrete network models. We establish the stability of the multiscale discrete network. Numerical results are presented for structured and random heterogeneous networks.

Interface problems, where distinct materials or physical domains meet, pose significant challenges in numerical simulations due to the discontinuities and sharp gradients across interfaces. Traditional finite element methods struggle to capture such behavior accurately. A new space transformed finite element method (ST-FEM) is developed for solving elliptic interface problems in $R^n$. A homeomorphic stretching transformation is introduced to obtain an equivalent problem in the transformed domain which can be solved easily, and the solution can be projected back to original domain by the inverse transformation. Compared with the existing methods, this new scheme has capability of handling discontinuities across the interface. The proposed approach has advantages in circumventing interface approximation properties and reducing the degree of freedom. We initially develop ST-FEM for elliptic problems and subsequently expand upon this concept to address elliptic interface problems. We prove optimal a priori error estimates in the $H^1$ and $L^2$ norms, and quasi-optimal error estimate for the maximum norm. Finally, numerical experiments demonstrate the superior accuracy and convergence properties of the ST-FEM when compared to the standard finite element method. The interface is assumed to be a $(n-1)$-sphere, nevertheless, our analysis can cover symmetric domains such as an ellipsoid or a cylinder.

We establish the exact expressions for the hypergeometric function of a Hermitian matrix argument. This result allows for the eigenvalues of the matrix argument to occur with arbitrary multiplicities and can be used for numerical computation. These exact expressions are particularly important since they provide the key ingredient which allows many results which involve this function to be useful from a practical engineering perspective.

In this article, we focus on an efficient multilevel finite element method to solve the time-dependent nonlinear Schrödinger equation which is one of the most important equations of mathematical physics. For the time derivative, we adopt implicit schemes including the backward Euler method and the Crank–Nicolson method. Based on these stable implicit schemes, the proposed method requires solving a nonlinear elliptic problem at each time step. For these nonlinear elliptic equations, a multilevel mesh sequence is constructed. At each mesh level, we first derive a rough approximation by correcting the approximation of the previous mesh level in a special correction subspace. The correction subspace is composed of a coarse finite element space and an additional approximate solution derived from the previous mesh level. Next, we only need to solve a linearized elliptic equation by inserting the rough approximation into the nonlinear term. Then, we derive an accurate approximate solution by performing the aforementioned solving process on the multilevel mesh sequence until we reach the final mesh level. Owing to the special construct of the correction subspace, we derive a multilevel finite element method to solve the nonlinear Schrödinger equation for the first time, and meanwhile we also derive an optimal error estimate with linear computational complexity. Additionally, unlike the existing multilevel methods for nonlinear problems, that typically require bounded second-order derivatives of the nonlinear terms, the nonlinear term in our study requires only one-order derivatives. Numerical results are provided to support our theoretical analysis and demonstrate the efficiency of the presented method.

In the past, the domain decomposition method was developed successfully for solving large-scale linear systems. However, the problems with significant nonlocal effect remain a major challenger for applying the method efficiently. In order to sort out the problem, a non-overlapping domain decomposition method with nonlocal interface boundary conditions was recently proposed and studied both theoretically and numerically. This paper is the report on the further development of the method, aiming to provide a comprehensive convergence analysis of the method, with supplementary numerical tests to support the theoretical result. The nonlocal effect of the problem is found to be reflected in both the governing equation and boundary conditions, and the effect of the latter was never taken into account, although playing a significant role in affecting the convergence. In addition, the paper extends the applicability of the analysis result drawn from the Poisson’s equation to more complicated problems by examining the symbols of the Steklov–Poincaré operators. The extended application includes a model equation arising from fluid dynamics and the high performance of the domain decomposition method in solving this equation is better elaborated.

With the development of multi-layer elastic systems in the field of engineering mechanics, the corresponding variational inequality theory and algorithm design have received more attention and research. In this study, a class of equivalent saddle point problems with interlayer Tresca friction conditions and the mixed finite element method are proposed and analyzed. Then, the convergence of the numerical solution of the mixed finite element method is theoretically proven, and the corresponding algebraic dual algorithm is provided. Finally, through numerical experiments, the mixed finite element method is not only compared with the layer decomposition method, but also its convergence relationship with respect to the spatial discretization parameter $H$ is verified.