Conservative space-time mesh refinement methods for the FDTD solution of Maxwell's equations

A new variational space-time mesh refinement method is proposed for the FDTD solution of Maxwell's equations. The main advantage of this method is to guarantee the conservation of a discrete energy that implies that the scheme remains L² stable under the usual CFL condition. The only additional cost induced by the mesh refinement is the inversion, at each time step, of a sparse symmetric positive definite linear system restricted to the unknowns located on the interface between coarse and fine grid. The method is presented in a rather general way and its stability is analyzed. An implementation is proposed for the Yee scheme. In this case, various numerical results in 3-D are presented in order to validate the approach and illustrate the practical interest of space-time mesh refinement methods.