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

Fariborz Maseeh Department of Mathematics and Statistics, Portland State University, Portland, OR, USA

^* Corresponding author: pazner@pdx.edu

Received: 3 December 2024
Accepted: 1 June 2025

Abstract

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 H¹-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.