Spectral methods, especially in their multidomain form, have become a key tool in scientific and engineering computation. This updated treatment retains the integration of theoretical and practical aspects from the original 1988 work while incorporating advancements in algorithms and theory. It offers a comprehensive overview of spectral methods for complex geometries, alongside detailed discussions of spectral algorithms for fluid dynamics in various geometries. The text introduces modern strategies for constructing spectral approximations in complex domains, including spectral elements, mortar elements, and discontinuous Galerkin methods, supported by numerous numerical examples. It also presents representative simulations from continuum mechanics and surveys efficient domain decomposition preconditioners suitable for parallel implementation. The focus on fluid dynamics in single domains covers established algorithms for boundary-layer equations, stability analyses, incompressible Navier-Stokes problems, and both inviscid and viscous compressible flows. Additionally, it provides an overview of modern techniques for computing incompressible flows in general geometries using high-order spectral discretizations. For reader convenience, essential concepts and formulas from a companion book on single domains are included.
