<p>We present and analyze a two-level restricted additive Schwarz (RAS) preconditioner for heterogeneous Helmholtz problems, based on a multiscale spectral generalized finite element method (MS-GFEM) proposed in [C. Ma, C. Alber, and R. Scheichl, SIAM. J. Numer. Anal., 61 (2023), pp. 1546–1584]. The preconditioner uses local solves with impedance boundary conditions, and a global coarse solve based on the MS-GFEM approximation space constructed from local eigenproblems. It is derived by first formulating MS-GFEM as a Richardson iterative method, and without using an oversampling technique, reduces to the preconditioner recently proposed and analyzed in [Q. Hu and Z.Li, arXiv 2402.06905]. We prove that both the Richardson iterative method and the preconditioner used within GMRES converge at a rate of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Λ</mi> </math></EquationSource> </InlineEquation> under some reasonable conditions, where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Λ</mi> </math></EquationSource> </InlineEquation> denotes the error of the underlying MS-GFEM approximation. Notably, the convergence proof of GMRES does not rely on the ‘Elman theory’. An exponential convergence property of MS-GFEM, resulting from oversampling, ensures that only a few iterations are needed for convergence with a small coarse space. In particular, in the constant-coefficient, non-trapping case, with <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$h\sim k^{-1-\gamma }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>h</mi> <mo>∼</mo> <msup> <mi>k</mi> <mrow> <mo>-</mo> <mn>1</mn> <mo>-</mo> <mi>γ</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> for some <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\gamma \in (0,1]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>, it holds that <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Lambda \sim k^{-1+\frac{\gamma }{2}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Λ</mi> <mo>∼</mo> <msup> <mi>k</mi> <mrow> <mo>-</mo> <mn>1</mn> <mo>+</mo> <mfrac> <mi>γ</mi> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>, with the coarse-space dimension <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\sim k^{d}\log ^{d}(k)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>∼</mo> <msup> <mi>k</mi> <mi>d</mi> </msup> <msup> <mo>log</mo> <mi>d</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We present extensive numerical experiments to illustrate the performance of the preconditioner, including 2D and 3D benchmark geophysics tests, and a high-contrast coefficient example arising in applications.</p>

Two-Level Restricted Additive Schwarz Preconditioner Based on Multiscale Spectral Generalized FEM for Heterogeneous Helmholtz Problems

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发表日期：

2025年11月22日

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Chupeng Ma,
Christian Alber,
Robert Scheichl,
Yongwei Zhang

摘要

We present and analyze a two-level restricted additive Schwarz (RAS) preconditioner for heterogeneous Helmholtz problems, based on a multiscale spectral generalized finite element method (MS-GFEM) proposed in [C. Ma, C. Alber, and R. Scheichl, SIAM. J. Numer. Anal., 61 (2023), pp. 1546–1584]. The preconditioner uses local solves with impedance boundary conditions, and a global coarse solve based on the MS-GFEM approximation space constructed from local eigenproblems. It is derived by first formulating MS-GFEM as a Richardson iterative method, and without using an oversampling technique, reduces to the preconditioner recently proposed and analyzed in [Q. Hu and Z.Li, arXiv 2402.06905]. We prove that both the Richardson iterative method and the preconditioner used within GMRES converge at a rate of $\Lambda $ $Λ$ under some reasonable conditions, where $\Lambda $ $Λ$ denotes the error of the underlying MS-GFEM approximation. Notably, the convergence proof of GMRES does not rely on the ‘Elman theory’. An exponential convergence property of MS-GFEM, resulting from oversampling, ensures that only a few iterations are needed for convergence with a small coarse space. In particular, in the constant-coefficient, non-trapping case, with $h\sim k^{-1-\gamma }$ $h \sim k^{- 1 - γ}$ for some $\gamma \in (0,1]$ $γ \in (0, 1]$ , it holds that $\Lambda \sim k^{-1+\frac{\gamma }{2}}$ $Λ \sim k^{- 1 + \frac{γ}{2}}$ , with the coarse-space dimension $\sim k^{d}\log ^{d}(k)$ $\sim k^{d} {log}^{d} (k)$ . We present extensive numerical experiments to illustrate the performance of the preconditioner, including 2D and 3D benchmark geophysics tests, and a high-contrast coefficient example arising in applications.