High-order spline finite element method for solving time-dependent electromagnetic waves

In this paper we propose a high-order spline finite element method for solving a class of time-dependent electromagnetic waves and its associated frequency-domain approach. A Fourier transform and its inverse are used for the time integration of the wave problem. The spatial discretization is performed using a partitioned mesh with tensorial spline functions to form bases of the discrete solution in the variational finite element space. Quadrature methods such as the Gauss-Hermite quadrature are implemented in the inverse Fourier transform to compute numerical solutions of the time-dependent electromagnetic waves. In the present study we carry out a rigorous convergence analysis and establish error estimates for the wave solution in the relevant norms. We also provide a full algorithmic description of the method and assess its performance by solving several test examples of time-dependent electromagnetic waves with known analytical solutions. The method is shown to verify the theoretical estimates and to provide highly accurate and efficient simulations. We also evaluate the computational performance of the proposed method for solving a problem of wave transmission through non-homogeneous materials. The obtained computational results confirm the excellent convergence, high accuracy and applicability of the proposed spline finite element method.