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Parameter-robust preconditioning for hybridizable symmetric discretizations

Esteban Henriquez, Jeonghun J. Lee, Sander Rhebergen

TL;DR

The paper tackles parameter-dependent linear systems arising from hybridizable discretizations by leveraging the Mardal–Winther framework to build a parameter-robust preconditioner for the full system and then analyzing conditions under which its reduced, statically condensed form remains robust. A central contribution is a lifting–spectral equivalence theorem that relates the full-system Schur complement to the reduced Schur complement, providing explicit conditions (including a uniform lifting bound) that guarantee robustness of S_P^{-1} for the reduced problem. The authors instantiate the theory for hybridizable discretizations of the Darcy and Stokes equations, deriving practical preconditioners and proving uniform bounds that are independent of mesh size and model parameters (e.g., $\xi$, $\gamma$, and $\nu$). Numerical experiments in 2D and 3D confirm parameter-robust convergence of the reduced preconditioners, including both exact and inexact variants, and demonstrate robustness under heterogeneous coefficients and classic flows like lid-driven cavities, indicating strong practical impact for large-scale, parameter-dependent simulations.

Abstract

Hybridizable discretizations allow for the elimination of local degrees-of-freedom leading to reduced linear systems. In this paper, we determine and analyse an approach to construct parameter-robust preconditioners for these reduced systems. Using the framework of Mardal and Winther (Numer. Linear Algebra Appl., 18(1):1--40, 2011) we first determine a parameter-robust preconditioner for the full system. We then eliminate the local degrees-of-freedom of this preconditioner to obtain a preconditioner for the reduced system. However, not all reduced preconditioners obtained in this way are automatically robust. We therefore present conditions that must be satisfied for the reduced preconditioner to be robust. To demonstrate our approach, we determine preconditioners for the reduced systems obtained from hybridizable discretizations of the Darcy and Stokes equations. Our analysis is verified by numerical examples in two and three dimensions.

Parameter-robust preconditioning for hybridizable symmetric discretizations

TL;DR

The paper tackles parameter-dependent linear systems arising from hybridizable discretizations by leveraging the Mardal–Winther framework to build a parameter-robust preconditioner for the full system and then analyzing conditions under which its reduced, statically condensed form remains robust. A central contribution is a lifting–spectral equivalence theorem that relates the full-system Schur complement to the reduced Schur complement, providing explicit conditions (including a uniform lifting bound) that guarantee robustness of S_P^{-1} for the reduced problem. The authors instantiate the theory for hybridizable discretizations of the Darcy and Stokes equations, deriving practical preconditioners and proving uniform bounds that are independent of mesh size and model parameters (e.g., , , and ). Numerical experiments in 2D and 3D confirm parameter-robust convergence of the reduced preconditioners, including both exact and inexact variants, and demonstrate robustness under heterogeneous coefficients and classic flows like lid-driven cavities, indicating strong practical impact for large-scale, parameter-dependent simulations.

Abstract

Hybridizable discretizations allow for the elimination of local degrees-of-freedom leading to reduced linear systems. In this paper, we determine and analyse an approach to construct parameter-robust preconditioners for these reduced systems. Using the framework of Mardal and Winther (Numer. Linear Algebra Appl., 18(1):1--40, 2011) we first determine a parameter-robust preconditioner for the full system. We then eliminate the local degrees-of-freedom of this preconditioner to obtain a preconditioner for the reduced system. However, not all reduced preconditioners obtained in this way are automatically robust. We therefore present conditions that must be satisfied for the reduced preconditioner to be robust. To demonstrate our approach, we determine preconditioners for the reduced systems obtained from hybridizable discretizations of the Darcy and Stokes equations. Our analysis is verified by numerical examples in two and three dimensions.

Paper Structure

This paper contains 28 sections, 14 theorems, 138 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Parameter-robust preconditioning
  3. Mardal--Winther preconditioning framework
  4. Preconditioning for a hybridizable discretization
  5. Application to the Darcy and Stokes problems
  6. Preliminaries
  7. The Darcy problem
  8. The Darcy problem: preconditioner $P^{-1}$
  9. The Darcy problem: the reduced preconditioner $S_P^{-1}$
  10. A counter example
  11. The Stokes problem
  12. The Stokes problem: preconditioner $P^{-1}$
  13. The Stokes problem: the reduced preconditioner $S_P^{-1}$
  14. Numerical Examples
  15. The Darcy problem: numerical results
  16. ...and 13 more sections

Key Result

Theorem 1

Let $A$ be the operator with the structure given in eq:wfgenop_block and let $P$ be the operator satisfying eq:defOpP with structure given in eq:Pblockstructure. Furthermore, let $S_A$ and $S_P$ be the operators defined in eq:SCop and eq:defSCSP, respectively. Assume that $A$ satisfies eq:wpconditio then Moreover, if eq:spectral-equivalence holds, then eq:liftingcondition holds with a uniform con

Theorems & Definitions (35)

  • Remark 1
  • Remark 2
  • Theorem 1
  • proof
  • Remark 3
  • Lemma 1
  • proof
  • Definition 1: Local solver for the Darcy problem
  • Lemma 2
  • Remark 4
  • ...and 25 more