Preconditioning for a pressure-robust HDG discretization of the Stokes equations

TL;DR

This paper presents a new preconditioner for a pressure-robust HDG discretization of the Stokes equations that reduces degrees-of-freedom and improves convergence and computational efficiency.

Contribution

A novel preconditioner for the fully condensed HDG Stokes system, with proven spectral equivalence and enhanced numerical performance over previous methods.

Findings

Fewer iterations needed for convergence.

Improved conservation properties in inexact solves.

Faster CPU times compared to previous preconditioners.

Abstract

We introduce a new preconditioner for a recently developed pressure-robust hybridized discontinuous Galerkin (HDG) finite element discretization of the Stokes equations. A feature of HDG methods is the straightforward elimination of degrees-of-freedom defined on the interior of an element. In our previous work (J. Sci. Comput., 77(3):1936--1952, 2018) we introduced a preconditioner for the case in which only the degrees-of-freedom associated with the element velocity were eliminated via static condensation. In this work we introduce a preconditioner for the statically condensed system in which the element pressure degrees-of-freedom are also eliminated. In doing so the number of globally coupled degrees-of-freedom are reduced, but at the expense of a more difficult problem to analyse. We will show, however, that the Schur complement of the statically condensed system is spectrally equivalent to a simple trace pressure mass matrix. This result is used to formulate a new, provably optimal preconditioner. Through numerical examples in two- and three-dimensions we show that the new preconditioned iterative method converges in fewer iterations, has superior conservation properties for inexact solves, and is faster in CPU time when compared to our previous preconditioner.