Abstract: The spectral element method has been used extensively for the simulation of fluid flows. The resulting linear systems are often not amenable to direct methods of solution, and are especially ill-conditioned. Domain decomposition preconditioners, well adapted to the solution on parallel computers, are proposed and analyzed; both two and three space dimensions are considered.

Second-order elliptic equations are considered first, and the now well-developed theory of domain decomposition methods for finite elements is fully extended to spectral elements. This includes an analysis of exotic coarse spaces, which have proven necessary for the efficient solution of elliptic problems with large discontinuities in the coefficients, as well as a study of overlapping methods. Estimates of the condition numbers of the Schur complement restricted to an edge (in two dimensions) or to a face (in three dimensions) are also given; in particular, a fast method is designed and studied in full detail for problems with many subregions.

The Stokes problem, when restricted to the space of discrete divergence free velocities, is symmetric positive definite. A number of preconditioners are proposed, which are based on previous results for the scalar elliptic case, and new global models. The construction of a basis for the constrained velocity space is not required,and the resulting condition numbers grow only weakly with the degree $N$ and are independent of the number of subdomains.

We also consider the stationary Navier-Stokes equations, solved with Newton's method. In each iteration, a non-symmetric indefinite problem is solved using a Schwarz preconditioner. A new coarse space is proposed which satisfies the usual properties required by the elliptic theory, and also a specific $H^1$-approximation property. The rate of convergence of the algorithm grows only weakly with $N$, and does not depend on the number of subdomains, or the Newton step.

Finally, a hierarchical basis preconditioner for the mortar finite element method in two dimensions is proposed and analyzed. It is also further shown that the analysis of the symmetric positive definite preconditioner can also be applied to construct preconditioners for symmetric indefinite problems arising from second-order elliptic equations. Numerical results are presented for the Helmholtz equation.