Fast Solvers and Domain Decomposition Preconditioners for Spectral Element Discretizations of Problems in H(curl) Bernhard Hientzsch TR2001-823 November 28, 2001 For problems with piecewise smooth solutions, spectral element methods hold great promise. They combine the exponential convergence of spectral methods with the geometric flexibility of finite elements. Spectral elements are well-established for scalar elliptic problems and problems of fluid dynamics, and recently the first methods for problems in H(curl) and H(div) were proposed. In this dissertation we study spectral element methods for a model problem. We first consider Maxwell's equation and derive the model problem in H(curl). Then we introduce anisotropic spectral Nédélec element discretizations with variable numerical integration for the model problem. We discuss their structure, and their convergence and approximation properties. We also obtain results on the norm of the Nédélec interpolants between Nédélec and Raviart-Thomas spaces of different degree, needed for the computation of the splitting constant for the domain decomposition preconditioner and the numerical analysis of nonlinear equations. We also prove a Friedrichs-like inequality for the model problem for the spectral case. We present fast direct solvers for the model problem on separable domains, taking advantage of the tensor product discretization and fast diagonalization methods. We use those fast solvers as local solvers in domain decomposition methods for problems that are too large to be solved directly, or posed on non-separable domains, and use them to compute and subassemble the Schur complement system corresponding to the interface. We also apply them in the direct solution of the Schur complement system for general domains. As an example for the domain decomposition methods that can be implemented with these tools, we introduce overlapping Schwarz methods, both one-level and two-level versions. We extend the theory for overlapping Schwarz methods to the spectral Nédélec element case. We reduce the proof of the condition number estimate to three basic estimates, and present theoretical and numerical results on those estimates. The technique of the proof works in both the two-dimensional and three-dimensional case. We also present numerical results for one-level and two-level methods in two dimensions.