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Unified analysis of saddle point problems via auxiliary space theory

Jongho Park

Abstract

We present sharp estimates for the extremal eigenvalues of the Schur complements arising in saddle point problems. These estimates are derived using the auxiliary space theory, in which a given iterative method is interpreted as an equivalent but more elementary iterative method on an auxiliary space, enabling us to obtain sharp convergence estimates. The proposed framework improves or refines several existing results, which can be recovered as corollaries of our results. To demonstrate the versatility of the framework, we present various applications from scientific computing: the augmented Lagrangian method, mixed finite element methods, and nonoverlapping domain decomposition methods. In all these applications, the condition numbers of the corresponding Schur complements can be estimated in a straightforward manner using the proposed framework.

Unified analysis of saddle point problems via auxiliary space theory

Abstract

We present sharp estimates for the extremal eigenvalues of the Schur complements arising in saddle point problems. These estimates are derived using the auxiliary space theory, in which a given iterative method is interpreted as an equivalent but more elementary iterative method on an auxiliary space, enabling us to obtain sharp convergence estimates. The proposed framework improves or refines several existing results, which can be recovered as corollaries of our results. To demonstrate the versatility of the framework, we present various applications from scientific computing: the augmented Lagrangian method, mixed finite element methods, and nonoverlapping domain decomposition methods. In all these applications, the condition numbers of the corresponding Schur complements can be estimated in a straightforward manner using the proposed framework.

Paper Structure

This paper contains 16 sections, 13 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Auxiliary space theory
  4. Saddle point problems
  5. Model problem
  6. Elimination of the primal variable
  7. Iterative methods
  8. Estimates for the Schur complements
  9. Application I: Augmented Lagrangian method
  10. Application II: Mixed finite element methods
  11. Darcy flow
  12. Stokes equations
  13. Application III: FETI and FETI-DP
  14. FETI
  15. FETI-DP
  16. ...and 1 more sections

Key Result

Lemma 2.1

Let $V$ and $\undertilde{V}$ be finite-dimensional vector spaces and let $\Pi \colon \undertilde{V} \to V$ be a surjective linear operator. Let $\undertilde{M} \colon \undertilde{V} \to \undertilde{V}$ be an SPD linear operator, and define Then $M$ is SPD. Moreover, it satisfies $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (25)

  • Lemma 2.1: auxiliary space lemma
  • Theorem 2.2
  • Proposition 3.1
  • Proof 1
  • Remark 3.2
  • Remark 3.3
  • Theorem 4.1
  • Proof 2
  • Corollary 4.2
  • Proof 3
  • ...and 15 more