In this paper linear elliptic problems with imperfect contact interface are considered, and a second order finite difference method is presented for linear problems, in which implicit jump condition are imposed on the interface. Then, the stability and convergence analysis of the FD scheme are given for the one-dimensional elliptic interface problem. Numerical examples are carried out for the elliptic problems with imperfect contact interfaces, and the results demonstrate that the scheme has second order accuracy for elliptic interface problems of implicit jump conditions with single and multiple imperfect interfaces.

In this paper, we study boundary element method with high order impedance boundary conditions (HOIBC) to solve Maxwell’s equations. The unknowns are electric and magnetic currents $\mathbf{J}$ and $\mathbf{M}.$ We propose several formulations and study the existence and uniqueness of the solution. Then, we discretize these formulations with a finite element method based on Lagrange elements. We give numerical tests of the HOIBC solution.

An H(curl)-conforming Nitsche extended finite element method is proposed for H(curl)-elliptic interface problems in three dimensional Lipschitz domains with smooth interfaces. Under interface-unfitted meshes, the continuous problems are discretized by an H(curl)-conforming extended finite element space, which is constructed based on the the lowest order of second family Nédélec edge elements (Whitney elements). A stabilization term defined on transmission faces is added to the standard discrete bilinear form. Stability results and the optimal error estimate in the parameter-dependent H(curl)-norm are derived, which are both uniform with respect to not only the mesh size and the interface position but also the physical parameters. Numerical experiments are carried out to validate theoretical results.

Novel time domain finite element methods are proposed to numerically solve the system of Maxwell’s equations with a cubic nonlinearity in the spatial 3D case. The effects of linear and nonlinear electric polarization are precisely modeled in this approach. In order to achieve an energy stable discretization at the semi-discrete and the fully discrete levels, a novel technique is developed to handle the discrete nonlinearity, with spatial discretization either using edge and face elements (Nédélec-Raviart-Thomas) or discontinuous spaces and edge elements (Lee-Madsen). In particular, the proposed time discretization scheme is unconditionally stable with respect to the electromagnetic energy and is free of any Courant-Friedrichs-Lewy-type condition. Optimal error estimates are presented at semi-discrete and fully discrete levels for the nonlinear problem. The methods are robust and allow for discretization of complicated geometries and nonlinearities of spatially 3D problems that can be directly derived from the full system of nonlinear Maxwell’s equations.

This paper studies the multiscale algorithm for parabolic integro-differential equations in composite media combining with Laplace transformation. The new contributions reported in this study are threefold: the convergence estimates with an explicit rate for the multiscale solutions of the equations in general domains are proved, the boundary layer solution is defined and the multiscale finite element algorithm which is suitable for parallel computation is presented. Numerical simulations are then carried out to validate the theoretical results.

In this article, we develop a Cartesian-mesh finite element method for solving Navier-Stokes interface problems with moving interfaces. The spatial discretization uses the immersed Crouzeix-Raviart nonconforming finite element introduced in [29]. A backward Euler full-discrete scheme is developed which embeds Newton’s iteration to treat the nonlinear convective term. The proposed IFE method does not require any stabilization terms while maintaining its convergence in optimal order. Numerical experiments with various interface shapes and jump coefficients are provided to demonstrate the accuracy of the proposed method. The numerical results are compared to the analytical solution as well as the standard finite element method with body-fitting meshes. Numerical results indicate the optimal order of convergence of the IFE method.

This article introduces a mesh-less, ray-based deep neural network method to solve the Helmholtz equation with high frequency. This method does not use an adaptive mesh refinement method, nor does it design a numerical scheme using some specially designed basis function to calculate the numerical solution, but it has the advantages of easy implementation and no mesh. We have carried out various numerical examples to prove the accuracy and efficiency of the proposed numerical method.