  This is the third part in a series on a mass conserving, high order, mixed
finite element method for Stokes flow. In this part, we study a block-diagonal
preconditioner for the indefinite Schur complement system arising from the
discretization of the Stokes equations using these elements. The underlying
finite element method is uniformly stable in both the mesh size h and
polynomial order p, and we prove bounds on the eigenvalues of the
preconditioned system which are independent of h and grow modestly in p. The
analysis relates the Schur complement system to an appropriate variational
setting with subspaces for which exact sequence properties and inf-sup
stability hold. Several numerical examples demonstrate agreement with the
theoretical results.
