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Analysis of the SORAS domain decomposition preconditioner for non-self-adjoint or indefinite problems

Marcella Bonazzoli, Xavier Claeys, Frédéric Nataf, Pierre-Henri Tournier

TL;DR

This work develops a general convergence framework for the one-level overlapping SORAS preconditioner applied to non-self-adjoint or indefinite systems, extending Helmholtz-type analyses to generic problems without requiring very fine coarse meshes. It establishes upper and lower bounds for GMRES convergence by bounding the norm of the preconditioned operator and the distance of its field of values from the origin, under verifiable assumptions tied to the local-global interface structure and partition-of-unity properties. The framework is applied to heterogeneous reaction-convection-diffusion equations with Robin-type transmissions, yielding explicit estimates for the local-to-global constants and demonstrating that increased subdomain overlap improves convergence, while highlighting the need for robust two-level strategies for certain coefficient regimes. Numerically, SORAS (and its non-symmetric variant ORAS) show favorable convergence behavior under adequate overlap and well-chosen transmission conditions, with results supporting the theoretical bounds and informing practical guidelines for overlap/partition design in non-SPD settings.

Abstract

We analyze the convergence of the one-level overlapping domain decomposition preconditioner SORAS (Symmetrized Optimized Restricted Additive Schwarz) applied to a generic linear system whose matrix is not necessarily symmetric/self-adjoint nor positive definite. By generalizing the theory for the Helmholtz equation developed in [I.G. Graham, E.A. Spence, and J. Zou, SIAM J.Numer.Anal., 2020], we identify a list of assumptions and estimates that are sufficient to obtain an upper bound on the norm of the preconditioned matrix, and a lower bound on the distance of its field of values from the origin. We stress that our theory is general in the sense that it is not specific to one particular boundary value problem. Moreover, it does not rely on a coarse mesh whose elements are sufficiently small. As an illustration of this framework, we prove new estimates for overlapping domain decomposition methods with Robin-type transmission conditions for the heterogeneous reaction-convection-diffusion equation (to prove the stability assumption for this equation we consider the case of a coercive bilinear form, which is non-symmetric, though).

Analysis of the SORAS domain decomposition preconditioner for non-self-adjoint or indefinite problems

TL;DR

This work develops a general convergence framework for the one-level overlapping SORAS preconditioner applied to non-self-adjoint or indefinite systems, extending Helmholtz-type analyses to generic problems without requiring very fine coarse meshes. It establishes upper and lower bounds for GMRES convergence by bounding the norm of the preconditioned operator and the distance of its field of values from the origin, under verifiable assumptions tied to the local-global interface structure and partition-of-unity properties. The framework is applied to heterogeneous reaction-convection-diffusion equations with Robin-type transmissions, yielding explicit estimates for the local-to-global constants and demonstrating that increased subdomain overlap improves convergence, while highlighting the need for robust two-level strategies for certain coefficient regimes. Numerically, SORAS (and its non-symmetric variant ORAS) show favorable convergence behavior under adequate overlap and well-chosen transmission conditions, with results supporting the theoretical bounds and informing practical guidelines for overlap/partition design in non-SPD settings.

Abstract

We analyze the convergence of the one-level overlapping domain decomposition preconditioner SORAS (Symmetrized Optimized Restricted Additive Schwarz) applied to a generic linear system whose matrix is not necessarily symmetric/self-adjoint nor positive definite. By generalizing the theory for the Helmholtz equation developed in [I.G. Graham, E.A. Spence, and J. Zou, SIAM J.Numer.Anal., 2020], we identify a list of assumptions and estimates that are sufficient to obtain an upper bound on the norm of the preconditioned matrix, and a lower bound on the distance of its field of values from the origin. We stress that our theory is general in the sense that it is not specific to one particular boundary value problem. Moreover, it does not rely on a coarse mesh whose elements are sufficiently small. As an illustration of this framework, we prove new estimates for overlapping domain decomposition methods with Robin-type transmission conditions for the heterogeneous reaction-convection-diffusion equation (to prove the stability assumption for this equation we consider the case of a coercive bilinear form, which is non-symmetric, though).

Paper Structure

This paper contains 8 sections, 12 theorems, 124 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Setting
  3. General theory
  4. Comments on the assumptions of Theorem \ref{['thm:main']}
  5. The reaction-convection-diffusion equation
  6. Estimates for the assumptions of Theorem \ref{['thm:main']}
  7. Summary of the constants
  8. Numerical experiments

Key Result

Theorem 3.1

For $j=1,\dots,N$, assume that for all global vectors of degrees of freedom $\mathbf{V} \in \mathbb{C}^n$ and local vectors of degrees of freedom $\mathbf{W}^j \in \mathbb{C}^{n_j}$ in $\Omega_j$ Suppose that there exists $\Lambda_0>0$ such that for all local vectors of degrees of freedom $\mathbf{W}^j \in \mathbb{C}^{n_j}$ in $\Omega_j$, $j=1,\dots,N$, we have and $\Lambda_1>0$ such that for all

Theorems & Definitions (26)

  • Theorem 3.1
  • Remark 3.2
  • proof
  • Remark 4.1
  • Definition 4.2: Characteristic length scale
  • Lemma 4.3: Multiplicative trace inequality, Grisvard:book:1985
  • Lemma 4.5: Continuity of the bilinear forms $a$ and $a_j$
  • proof
  • Lemma 4.6: Coercivity of the bilinear forms $a$ and $a_j$
  • proof
  • ...and 16 more