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Schwarz preconditioner with $H_k$-GenEO coarse space for the indefinite Helmholtz problem

Victorita Dolean, Mark Fry, Ivan G. Graham, Matthias Langer

TL;DR

The paper addresses solving the indefinite Helmholtz problem with highly variable coefficients using two-level domain decomposition. It introduces the $H_k$-GenEO coarse space, built directly from the full indefinite operator and a $k$-weighted inner product, to achieve robustness as the wave-number $k$ increases. The authors prove $k$-explicit bounds on the fine-grid diameter $H^f$ and eigenvalue tolerance $\tau$ that guarantee GMRES convergence independent of the number of subdomains and coefficient variations, and they provide stability and solvability results for the associated local and coarse operators. This yields a scalable, robust preconditioner for high-frequency Helmholtz problems, with practical guidance on parameter choices in terms of $k$, $H^f$, and $\tau$.

Abstract

GenEO (`Generalised Eigenvalue problems on the Overlap') is a method from the family of spectral coarse spaces that can efficiently rely on local eigensolves in order to build a robust parallel domain decomposition preconditioner for elliptic PDEs. When used as a preconditioner in a conjugate gradient, this method is extremely efficient in the positive-definite case, yielding an iteration count completely independent of the number of subdomains and heterogeneity. In a previous work this theory was extended to the cased of convection--diffusion--reaction problems, which may be non-self-adjoint and indefinite, and whose discretisations are solved with preconditioned GMRES. The GenEO coarse space was then defined here using a generalised eigenvalue problem based on a self-adjoint and positive definite subproblem. The resulting method, called $Δ$-GenEO becomes robust with respect to the variation of the coefficient of the diffusion term in the operator and depends only very mildly on variations of the other coefficients. However, the iteration number estimates get worse as the non-self-adjointness and indefiniteness of the operator increases, which is often the case for the high frequency Helmholtz problems. In this work, we will improve on this aspect by introducing a new version, called $H_k$-GenEO, which uses a generalised eigenvalue problem based directly on the indefinite operator which will lead to a robust method with respect to the increase in the wave-number. We provide theoretical estimates showing the dependence of the size of the coarse space on the wave-number.

Schwarz preconditioner with $H_k$-GenEO coarse space for the indefinite Helmholtz problem

TL;DR

The paper addresses solving the indefinite Helmholtz problem with highly variable coefficients using two-level domain decomposition. It introduces the -GenEO coarse space, built directly from the full indefinite operator and a -weighted inner product, to achieve robustness as the wave-number increases. The authors prove -explicit bounds on the fine-grid diameter and eigenvalue tolerance that guarantee GMRES convergence independent of the number of subdomains and coefficient variations, and they provide stability and solvability results for the associated local and coarse operators. This yields a scalable, robust preconditioner for high-frequency Helmholtz problems, with practical guidance on parameter choices in terms of , , and .

Abstract

GenEO (`Generalised Eigenvalue problems on the Overlap') is a method from the family of spectral coarse spaces that can efficiently rely on local eigensolves in order to build a robust parallel domain decomposition preconditioner for elliptic PDEs. When used as a preconditioner in a conjugate gradient, this method is extremely efficient in the positive-definite case, yielding an iteration count completely independent of the number of subdomains and heterogeneity. In a previous work this theory was extended to the cased of convection--diffusion--reaction problems, which may be non-self-adjoint and indefinite, and whose discretisations are solved with preconditioned GMRES. The GenEO coarse space was then defined here using a generalised eigenvalue problem based on a self-adjoint and positive definite subproblem. The resulting method, called -GenEO becomes robust with respect to the variation of the coefficient of the diffusion term in the operator and depends only very mildly on variations of the other coefficients. However, the iteration number estimates get worse as the non-self-adjointness and indefiniteness of the operator increases, which is often the case for the high frequency Helmholtz problems. In this work, we will improve on this aspect by introducing a new version, called -GenEO, which uses a generalised eigenvalue problem based directly on the indefinite operator which will lead to a robust method with respect to the increase in the wave-number. We provide theoretical estimates showing the dependence of the size of the coarse space on the wave-number.
Paper Structure (11 sections, 17 theorems, 145 equations)

This paper contains 11 sections, 17 theorems, 145 equations.

Table of Contents

  1. Introduction
  2. Useful Background
  3. Problem formulation and discretisation
  4. Abstract spectral theory for the generalised indefinite eigenvalue problems
  5. Domain Decomposition
  6. The $H_k$-GenEO coarse space
  7. Statement of the main result and theoretical tools
  8. Properties of the $H_k$-GenEO coarse space
  9. Solvability and stability of $T^f_{i,j}$, $i=1,\ldots,N$ and $j=1,\ldots,Q$
  10. Existence and stability of $T_0$
  11. Proof of the main result

Key Result

Lemma 2.4

Let Assumptions ass: 2_1 and Ass: 2_3 hold. Then there exists an $h_0>0$ such that, for each $h$ with $0 < h < h_0$, the problem eq: 2_11 has a unique solution $u_h \in V_H$. Moreover, let $u$ be the unique solution of weak_form. Then, for every $\varepsilon>0$ there exists $h_1=h_1(\varepsilon)>0$ and

Theorems & Definitions (40)

  • Lemma 2.4: Schatz and Wang, 1996
  • Lemma 2.5: Friedrichs inequality
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • proof
  • Definition 2.8
  • Remark 2.10
  • Proposition 2.11
  • proof
  • ...and 30 more