Subspace and auxiliary space preconditioners for high-order interior penalty discretizations in $H(\mathrm{div})$

TL;DR

This paper develops and analyzes three robust preconditioners for high-order interior penalty discretizations in $H(\mathrm{div})$, improving solver efficiency for divergence-free discretizations in fluid dynamics problems.

Contribution

The paper introduces and rigorously analyzes three new preconditioners for high-order $H(\mathrm{div})$ discretizations, demonstrating their robustness and effectiveness across various mesh types and polynomial degrees.

Findings

Preconditioners are well-conditioned independently of mesh size, polynomial degree, and penalty parameter.

All three preconditioners are robust with respect to mesh size on general meshes.

Numerical results show mild growth of iteration counts with increasing polynomial degree.

Abstract

In this paper, we construct and analyze preconditioners for the interior penalty discontinuous Galerkin discretization posed in the space $H(\mathrm{div})$. 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 $\eta$. 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.