Preface

For a long time, wave equations in general and Maxwell’s equations in particular were solved in the time-harmonic domain by finite element methods (FEM). Equations in time domain were solved by using second-order finite difference methods (FDM) whose outstanding representative is the Yee scheme (also called FDTD) used for electromagnetic waves since 1966 and still alive. Unfortunately, second-order FDM found their limits in modeling long time propagation which often occur in physical problems, either because the wavelength is small compared to the size of the efficient domain, or because we have trapped waves in cavities. Actually, the number of points required to get an accurate solution grows with the interval of resolution in time. A palliative to this drawback seemed to be the use of higher order FDM which enable us to increase the size of the space-step while keeping a satisfactory accuracy. However, this technique is very troublesome to model complex geometries because of the large size of the grid cells. People were nevertheless reluctant to use FEM in the time domain (called FETD in the electromagnetic community), which could ensure a good approximation of these geometries. The reason came from the presence of a mass matrix which is naturally diagonal for FDM, but n-diagonal for FEM, with n increasing with the dimension of space and the order of the method. This matrix requires to be inverted at each time-step and substantially slows down the performance of the method, even when using iterative algorithms of inversion. An answer to this difficult problem was given in two ways. A first way, introduced by Cohen et al. [1] for wave equations, was based on mass lumping of FEM on quadrilateral and hexahedral meshes with Gauss–Lobatto points. Actually, this idea was first used for reservoir simulation [2] and neutronics [3] and was later