Subspace and auxiliary space preconditioners for high-order interior penalty discretizations in <i>H</i>(div)

In this paper, we construct and analyze preconditioners for the interior penalty discontinuous Galerkin discretization posed in the space H (div). These discretizations are used as one component in exactly divergence-free pressure-robust discretizations for the Stokes problem. Three preconditioners are presently considered: a subspace correction preconditioner using vertex patches and the lowest-order H 1 -conforming space as a coarse space, a fictitious space preconditioner using the degree- p discontinuous Galerkin space, and an auxiliary space preconditioner using the degree-( p − 1) discontinuous Galerkin space and a block Jacobi smoother. On certain classes of meshes, the subspace and fictitious space preconditioners result in provably well-conditioned systems, independent of the mesh size h , polynomial degree p , and penalty parameter η . All three preconditioners are shown to be robust with respect to h on general meshes, and numerical results indicate that the iteration counts grow only mildly with respect to p in the general case. Numerical examples illustrate the convergence properties of the preconditioners applied to structured and unstructured meshes. These solvers are used to construct block-diagonal preconditioners for the Stokes problem, which result in uniform convergence when used with MINRES.