We introduce a preconditioning technique based on Domain Decomposition and the Fast Diagonalization Method that can be applied to tensor product based discretizations of the steady convection-diffusion and the linearized Navier-Stokes equations. The method is based on iterative substructuring where fast diagonalization is used to efficiently eliminate the interior degrees of freedom and subsidiary subdomain solves. We demonstrate the effectiveness of this preconditioner in numerical simulations using a spectral element discretization.

This work extends the use of Fast Diagonalization to steady convection-diffusion systems. We also extend the “least-squares commutator” preconditioner, originally developed for the finite element method [19], to a matrix-free spectral element framework. We show that these two advances, when used together, allow for efficient computation of steady-state solutions the incompressible Navier-Stokes equations using high-order spectral element discretizations.