Finite Element and Discontinuous Galerkin Methods for Transient Wave Equations

This monograph presents methods for time domain wave propagation problems. It also describes how to implement them in physical models. Highlights include the construction of 3D finite elements on non classical elementary shapes such as prisms or pyramids, an extensive analysis of discontinuous Galerkin methods, and coverage of Maxwell's equations, which includes an extensive treatment of the delicate question of spurious modes (specific to electromagnetic waves). After recalling such classical models as acoustics, linear elastodynamics, and electromagnetism, the authors present a wide variety of finite elements of different shapes useful for wave equations. They then go on to focus on the construction of efficient continuous and discontinuous Galerkin methods and study their accuracy by plane wave techniques and a priori error estimates. In addition, the volume analyzes approximation of unbounded domains and deals with time approximation including local time-stepping and some complex models. Throughout, emphasis is put on the accuracy and computational efficiency of the presented numerical methods, with attention brought to their practical aspects. Coverage also details the theoretical foundations and numerical analysis of these methods. As a result, this monograph will be of interest to practitioners, researchers, engineers, and graduate students involved in the numerical simulation of waves.