In this paper we propose a Deep Learning Galerkin Method (DGM) based on the deep neural network learning algorithm to approximate the general second-order linear elliptic problem. This method is a combination of Galerkin Method and machine learning. The DGM uses the deep neural network instead of the linear combination of basis functions. Our algorithm is meshfree and we train the neural network by randomly sampling the space points and using the gradient descent algorithm to satisfy the differential operators and boundary conditions. Moreover, the approximate ability of a neural networks' solution to the exact solution is proved by the convergence of the loss function and the convergence of the neural network to the exact solution in $L^2$ norm under certain conditions. Finally, some numerical experiments reflect the approximation ability of the neural networks intuitively.

This paper presents an efficient method for the computation of two highly oscillatory integrals having logarithmic and Cauchy-logarithmic singularities. This approach first requires the transformation of the original oscillatory integrals into a sum of line integrals with semi-infinite intervals. Afterwards, the coefficients of the three-term recurrence relation that satisfy the orthogonal polynomial are obtained by using the method based on moments, where classical Laguerre and Gautschi's logarithmic weight functions are employed. The algorithm reveals that with fixed $n$, the method is capable of achieving significant figures within a short time. Furthermore, the approach yields higher accuracy as the frequency increases. The results of numerical experiments are given to substantiate our theoretical analysis.

In recent years, fractional differential equations have been extensively applied to model various complex dynamic systems. The studies on highly accurate and efficient numerical methods for fractional differential equations have become necessary. In this paper, an effective recurrence algorithm for computing both the fractional Riemann-Liouville and Caputo derivatives is proposed, and then spectral collocation methods based on the algorithm are investigated for solving fractional differential equations. By the recurrence method, the numerical stability with respect to $N$, the number of collocation points, can be improved remarkably in comparison with direct algorithm. Its robustness ensures that a highly accurate spectral collocation method can be applied widely to various fractional differential equations.

We consider a second order singularly perturbed boundary value problem, of reaction-convection-diffusion type with two small parameters, and the approximation of its solution by the $hp$ version of the Finite Element Method on the so-called $Spectral$ $Boundary$ $Layer$ mesh. We show that the method converges uniformly, with respect to both singular perturbation parameters, at an exponential rate when the error is measured in the energy norm. Numerical examples are also presented, which illustrate our theoretical findings as well as compare the proposed method with others found in the literature.

A modified primal-dual weak Galerkin (M-PDWG) finite element method is designed for the second order elliptic equation in non-divergence form. Compared with the existing PDWG methods proposed in [6], the system of equations resulting from the M-PDWG scheme could be equivalently simplified into one equation involving only the primal variable by eliminating the dual variable (Lagrange multiplier). The resulting simplified system thus has significantly fewer degrees of freedom than the one resulting from existing PDWG scheme. Optimal order error estimates are derived for the numerical approximations in the discrete $H^2$-norm, $H^1$-norm and $L^2$-norm respectively. Extensive numerical results are demonstrated for both the smooth and non-smooth coefficients on convex and non-convex domains to verify the accuracy of the theory developed in this paper.

In this paper, we develop and analyze finite difference methods for the 3D Maxwell's equations in the time domain in three different types of linear dispersive media described as Debye, Lorentz and cold plasma. These methods are constructed by extending the Yee-Finite Difference Time Domain (FDTD) method to linear dispersive materials. We analyze the stability criterion for the FDTD schemes by using the energy method. Based on energy identities for the continuous models, we derive discrete energy estimates for the FDTD schemes for the three dispersive models. We also prove the convergence of the FDTD schemes with perfect electric conducting boundary conditions, which describes the second order accuracy of the methods in both time and space. The discrete divergence-free conditions of the FDTD schemes are studied. Lastly, numerical examples are given to demonstrate and confirm our results.