Modeling and an immersed finite element method for an interface wave equation

The electromagnetic field, which is governed by Maxwell's equation, plays a key role in plasma simulation. In this article, we first derive the interface conditions when we rewrite the interface Maxwell's equation, whose problem domain involves complex media such as objects of different materials, into a parabolic-hyperbolic type of interface model, which is a modified wave equation by adding a first order time derivative term due to the lossy medium. Based on the interface conditions and the existing bilinear immersed finite element space for the interface Poisson equation, we propose an immersed finite element method for the spatial discretization of this parabolic-hyperbolic interface equation on a Cartesian mesh independent of the interface. Then we use a second order finite difference method for the temporal discretization in order to develop the full discretization scheme. Compared with the unstructured body-fitting mesh which is needed by the traditional finite element method for interface problems, the Cartesian mesh independent of the interface will significantly benefit the plasma simulation in the electromagnetic field, especially the particle models (such as the particle-in-cell method). This method provides a new efficient way to study varying electromagnetic field with objects of different materials on a Cartesian mesh independent of the interface, hence builds a solid foundation for the further study on the motion of plasma in this electromagnetic field. Numerical examples are provided to demonstrate the features of the proposed method.