Spectral Methods. Evolution to Complex Geometries and Applications to Fluid Dynamics

Spectral methods, particularly in their multidomain version, have become firmly established as a mainstream tool for scientific and engineering computation. This book provides an extensive and critical overview of the essential algorithmic and theoretical aspects of spectral methods for complex geometries, in addition to detailed discussions of spectral algorithms for fluid dynamics in simple and complex geometries. Modern strategies for constructing spectral approximations in complex domains, such as spectral elements, mortar elements, and discontinuous Galerkin methods, as well as patching collocation, are introduced, analyzed in their mathematical aspects, and demonstrated by means of numerous numerical examples. Efficient domain decomposition preconditioners (of both Schwarz and Schur type) that are amenable to parallel implementation are deeply investigated. Representative simulations from continuum mechanics are also shown.