Spatiotemporal mesh refining for Maxwell equations

This work deals with the simulation of waves propagation in complex medium. Our aim is the following one. On most of the propagation domain, we use traditional techniques, typically the Yee's scheme, to discretize Maxwell's equations, while, near the obstacle, we use a finer local grid in order to take better account of geometrical details. We are then led to take a local time step in order to avoid the increase of numerical dispersion in the principal grid. The difficulty is then to find a good scheme to connect the two grids. First, we make a mathematical and numerical study on the method classically employed in the electromagnetic literature using space-time interpolations on the interface between the two grids. Our study points out a new kind of instability with this type of approach. In a second part, we present a new strategy based on weak continuity relations at the interface that ensures stability via the conservation of a discrete energy. In the third part, we show that this scheme enters a more general class of variational problems. The scheme for gluing the two grids is related to the mortar element method and is carried out via the introduction of a Lagrange parameter defined on the interface. This generalization enables us to carry out more easily the space-time mesh refinement for Maxwell's equations D. (author)