On Maxwell Equations with the Transparent Boundary Condition

Here D ⊂ R is a bounded domain with Lipschitz boundary ΓD, E is the electric field, H is the magnetic field, x̂ = x/|x|, and n is the unit outer normal to ΓD. The applied current J and the initial conditions E0,H0 are assumed to be supported in the circle BR = {x ∈ R : |x| < R} for some R > 0. The electric permittivity ε and magnetic permeability μ are assumed to be positive constants. We remark that the results in this paper can be easily extended to solve scattering problems with other boundary conditions such as the impedance boundary condition on ΓD. One of the fundamental problems in the efficient simulation of the wave propagation is the reduction of the exterior problem which is defined in the unbounded domain to the problem in the bounded domain. The transparent boundary condition plays an important role in the construction of various approximate absorbing boundary conditions for the simulation of wave