For problems with piecewise smooth solutions, spectral element methods (SEM) hold great promise, combining the fast exponential convergence of spectral methods with the flexibility of finite elements. Once the problems are discretized with SEM, fast direct solvers are needed for the resulting systems. When the systems become too large, iterative solvers and good preconditioners are needed. Up to now, SEM have been mostly applied to elliptic problems (e.g., Poisson equations) and to fluid dynamics. This book applies SEM to problems derived from Maxwell equations, which present new issues and challenges. We derive and describe the problems, discretize them, develop several fast solvers for them, construct domain decomposition preconditioners for them, test the solvers and preconditioners on examples, and prove that the preconditioners are optimal, introducing many ideas, methods, and results along the way. The book presents an accessible introduction to SEM, domain decomposition methods, and fast solvers, applied to computational electromagnetics. It should be useful and interesting for both beginners and specialists.

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