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Accelerating non-local exchange in generalized optimized Schwarz methods

Roxane Atchekzai, Xavier Claeys

TL;DR

The paper addresses accelerating the non-local exchange in generalized optimised Schwarz methods for time-harmonic waves. It introduces an efficient, provably accurate approach that combines a recycling strategy with truncated preconditioned conjugate gradient solves to approximate the non-local exchange operator, preserving the coercive skeleton formulation and overall convergence. Theoretical analysis shows geometric convergence under a computable bound on the PCG error, and numerical experiments demonstrate substantial speedups with minimal impact on accuracy, enabling scalable domain-decomposition solvers for high-frequency problems with cross-points.

Abstract

The generalized optimised Schwarz method proposed in [Claeys & Parolin, 2022] is a variant of the Després algorithm for solving harmonic wave problems where transmission conditions are enforced by means of a non-local exchange operator. We introduce and analyse an acceleration technique that significantly reduces the cost of applying this exchange operator without deteriorating the precision and convergence speed of the overall domain decomposition algorithm.

Accelerating non-local exchange in generalized optimized Schwarz methods

TL;DR

The paper addresses accelerating the non-local exchange in generalized optimised Schwarz methods for time-harmonic waves. It introduces an efficient, provably accurate approach that combines a recycling strategy with truncated preconditioned conjugate gradient solves to approximate the non-local exchange operator, preserving the coercive skeleton formulation and overall convergence. Theoretical analysis shows geometric convergence under a computable bound on the PCG error, and numerical experiments demonstrate substantial speedups with minimal impact on accuracy, enabling scalable domain-decomposition solvers for high-frequency problems with cross-points.

Abstract

The generalized optimised Schwarz method proposed in [Claeys & Parolin, 2022] is a variant of the Després algorithm for solving harmonic wave problems where transmission conditions are enforced by means of a non-local exchange operator. We introduce and analyse an acceleration technique that significantly reduces the cost of applying this exchange operator without deteriorating the precision and convergence speed of the overall domain decomposition algorithm.
Paper Structure (7 sections, 4 theorems, 35 equations, 3 figures)

This paper contains 7 sections, 4 theorems, 35 equations, 3 figures.

Table of Contents

  1. Scattering problem under study
  2. Decomposition of the computational domain
  3. Exchange operator
  4. Skeleton formulation
  5. Approximation of the exchange operator
  6. Numerical experiment
  7. Convergence analysis

Key Result

Proposition 4.1

The tuple function $\boldsymbol{u}\in\mathbb{X}_{h}(\Omega)$ solves SecondDiscreteFormulation if and only if there exists $\boldsymbol{q}\in \mathbb{V}_{h}(\Sigma)^*$ satisfying $\mathrm{B}^*\boldsymbol{q} = (\mathrm{A}-i\mathrm{B}^*\mathrm{T}\mathrm{B})\boldsymbol{u} - \boldsymbol{l}$ and the skele

Figures (3)

  • Figure 1: Left: Computational domain. Right: Real part of the reference solution.
  • Figure 2: Relative error $\Vert \boldsymbol{q}^{(\infty)} - \tilde{\boldsymbol{q}}^{(n)}\Vert_{\mathrm{T}^{-1}}/\Vert \boldsymbol{q}^{(\infty)}\Vert_{\mathrm{T}^{-1}}$ versus iteration number $n$ in Richardson's Algorithm. No initial guess of PCG i.e. no recycling strategy. Left: no limitation on PCG iteration count i.e. $k_{\textsc{max}} = \infty$. Right: imposing in addition $k\leq k_{\textsc{max}}$.
  • Figure 3: Relative error $\Vert \boldsymbol{q}^{(\infty)} - \tilde{\boldsymbol{q}}^{(n)}\Vert_{\mathrm{T}^{-1}}/\Vert \boldsymbol{q}^{(\infty)}\Vert_{\mathrm{T}^{-1}}$ versus iteration number $n$ in Richardson's Algorithm with a recycled initial guess for PCG and $k\leq k_{\textsc{max}} = 5,10$.

Theorems & Definitions (4)

  • Proposition 4.1
  • Proposition 4.2
  • Lemma 7.1
  • Lemma 7.2