A combination of block-Jacobi and deflation preconditioning is used to
solve a high-order discontinuous element-based collocation discretization
of the Schur complement of the Poisson-Neumann system as arises in the
operator splitting of the incompressible Navier-Stokes equations.
The ill-posedness of the Poisson-Neumann system manifests as an
inconsistency of the Schur complement problem, but it is shown that this
can be accounted for with appropriate projections out of the null space
of the Schur complement matrix without affecting the accuracy of the
solution. The block-Jacobi preconditioner, combined with deflation, is
shown to yield GMRES convergence independent of the polynomial order of
expansion within an element. Finally, while the number of GMRES
iterations does grow as the element size is reduced (e.g.
-refinement), the dependence is very mild; the number of GMRES
iterations roughly doubles as the element size is divided by a factor of
six. In light of these numerical results, the deflated Schur complement
approach seems practicable, especially for high-order methods given its
convergence independent of polynomial order.