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Preconditioning a hybridizable discontinuous Galerkin method for Navier-Stokes at high Reynolds number

Alexander D. Lindsay, Sander Rhebergen, Ben S. Southworth

TL;DR

The paper develops an augmented Lagrangian–type, divergence-conforming preconditioner for a hybridizable discontinuous Galerkin discretization of the stationary Navier–Stokes equations in advection-dominated regimes. By focusing on the trace pressure Schur complement and employing a mass-matrix–based approximation, the authors derive practical AL-like and static-condensation–aware preconditioners and couple them with a STRUMPACK multifrontal LU solver for the augmented velocity block. The approach demonstrates robustness across mesh refinements and Reynolds numbers in canonical 2D tests (lid-driven cavity and backward-facing step), with detailed solver performance and scalability analysis. The work provides a fully algebraic, black-box solver framework suitable for HDG-based CFD simulations, including extensions to EDG-HDG and potential applicability to turbulence modeling contexts.

Abstract

We introduce a preconditioner for a hybridizable discontinuous Galerkin discretization of the linearized Navier-Stokes equations at high Reynolds number. The preconditioner is based on an augmented Lagrangian approach of the full discretization. Unlike standard grad-div type augmentation, however, we consider augmentation based on divergence-conformity. With this augmentation we introduce two different, well-conditioned, and easy to solve matrices to approximate the trace pressure Schur complement. To introduce a completely algebraic solver, we propose to use multifrontal sparse LU solvers using butterfly compression to solve the trace velocity block. Numerical examples demonstrate that the trace pressure Schur complement is highly robust in mesh spacing and Reynolds number and that the multifrontal inexact LU performs well for a wide range of Reynolds numbers.

Preconditioning a hybridizable discontinuous Galerkin method for Navier-Stokes at high Reynolds number

TL;DR

The paper develops an augmented Lagrangian–type, divergence-conforming preconditioner for a hybridizable discontinuous Galerkin discretization of the stationary Navier–Stokes equations in advection-dominated regimes. By focusing on the trace pressure Schur complement and employing a mass-matrix–based approximation, the authors derive practical AL-like and static-condensation–aware preconditioners and couple them with a STRUMPACK multifrontal LU solver for the augmented velocity block. The approach demonstrates robustness across mesh refinements and Reynolds numbers in canonical 2D tests (lid-driven cavity and backward-facing step), with detailed solver performance and scalability analysis. The work provides a fully algebraic, black-box solver framework suitable for HDG-based CFD simulations, including extensions to EDG-HDG and potential applicability to turbulence modeling contexts.

Abstract

We introduce a preconditioner for a hybridizable discontinuous Galerkin discretization of the linearized Navier-Stokes equations at high Reynolds number. The preconditioner is based on an augmented Lagrangian approach of the full discretization. Unlike standard grad-div type augmentation, however, we consider augmentation based on divergence-conformity. With this augmentation we introduce two different, well-conditioned, and easy to solve matrices to approximate the trace pressure Schur complement. To introduce a completely algebraic solver, we propose to use multifrontal sparse LU solvers using butterfly compression to solve the trace velocity block. Numerical examples demonstrate that the trace pressure Schur complement is highly robust in mesh spacing and Reynolds number and that the multifrontal inexact LU performs well for a wide range of Reynolds numbers.

Paper Structure

This paper contains 19 sections, 6 theorems, 76 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. HDG for the Navier--Stokes equations
  3. Notation and finite element spaces
  4. The discretization
  5. The algebraic formulation
  6. Preconditioning
  7. On the inverse of singular perturbations
  8. Preconditioning for HDG
  9. AL-like preconditioning
  10. Preconditioner for static condensation
  11. Solving the augmented velocity block
  12. Numerical examples
  13. Test cases
  14. Results
  15. Further discussion
  16. ...and 4 more sections

Key Result

Lemma 1

Let $A$ be a nonsingular $n \times n$ matrix, $J \ne 0$ be a symmetric, singular $n \times n$ matrix, and $B$ be an $m \times n$ matrix ($m < n$) such that $\mathcal{N}(J)\subseteq \mathcal{N}(B)$ and $\mathcal{R}(B^T) = \mathcal{R}(J)$. Let $\gamma>0$ be a scalar constant and assume the following m Let $P$ be an orthogonal projection onto $\mathcal{N}(J)$, let $Q=I-P$ denote its complement, and d

Figures (2)

  • Figure 1: Solver diagram.
  • Figure 2: Pressure colormaps plus streamline plots of the lid-driven cavity and backward-facing step problems with $\text{Re} = 10000.0$

Theorems & Definitions (13)

  • Lemma 1: Singular perturbation of saddle-point systems
  • proof
  • Corollary 1: Augmented Lagrangian Schur complements
  • proof
  • Lemma 2
  • proof
  • Remark 1
  • Lemma 3: inf-sup condition
  • proof
  • Lemma 4: Well-posedness
  • ...and 3 more