[Paper Review] Domain Decomposition Methods based on quasi-optimal transmission operators for the solution of Helmholtz transmission problems

Yassine Boubendir, Carlos Jerez-Hanckes+2 more|arXiv (Cornell University)|Oct 7, 2017

Electromagnetic Scattering and Analysis22 references3 citations

TL;DR

This paper presents non-overlapping domain decomposition methods (DDM) for Helmholtz transmission problems using quasi-optimal transmission operators derived from approximations of Dirichlet-to-Neumann (DtN) operators. By employing complexified hypersingular boundary integral operators or square root Fourier multipliers, the method achieves robust convergence in few iterations—even in high-contrast, high-frequency regimes—with minimal sensitivity to frequency, contrast, or number of subdomains.

ABSTRACT

We present non-overlapping Domain Decomposition Methods (DDM) based on quasi-optimal transmission operators for the solution of Helmholtz transmission problems with piece-wise constant material properties. The quasi-optimal transmission boundary conditions incorporate readily available approximations of Dirichlet-to-Neumann operators. These approximations consist of either complexified hypersingular boundary integral operators for the Helmholtz equation or square root Fourier multipliers with complex wavenumbers. We show that under certain regularity assumptions on the closed interface of material discontinuity, the DDM with quasi-optimal transmission conditions are well-posed. We present a DDM framework based on Robin-to-Robin (RtR) operators that can be computed robustly via boundary integral formulations. More importantly, the use of quasi-optimal transmission operators results in DDM that converge in small numbers of iterations even in the challenging high-contrast, high-frequency regime of Helmholtz transmission problems. Furthermore, the DDM presented in this text require only minor modifications to handle the case of transmission problems in partially coated domains, while still maintaining excellent convergence properties. We also investigate the dependence of the DDM iterative performance on the number of subdomains.

Motivation & Objective

• Address the slow convergence of traditional domain decomposition methods (DDM) for Helmholtz transmission problems in high-contrast, high-frequency regimes.
• Overcome limitations of classical Robin boundary conditions and standard boundary integral equation (BIE) formulations that require many Krylov iterations in challenging parameter regimes.
• Develop a DDM framework that maintains robust convergence for complex material configurations, including partially coated domains.
• Ensure well-posedness of subdomain problems through complexification of wavenumbers in transmission operators.
• Minimize iteration counts across varying numbers of subdomains and high-frequency parameters, enabling scalable and efficient solution strategies.

Proposed method

• Formulate a non-overlapping DDM using quasi-optimal transmission operators based on approximations of Dirichlet-to-Neumann (DtN) operators.
• Use complexified hypersingular boundary integral operators (BIOs) or square root Fourier multipliers with complex wavenumbers as transmission operators.
• Ensure well-posedness of subdomain Helmholtz problems by complexifying wavenumbers in transmission operators to satisfy coercivity conditions.
• Implement a Robin-to-Robin (RtR) operator framework computed via boundary integral formulations for robust and efficient iteration.
• Adapt the method to partially coated domains by modifying transmission operators without sacrificing convergence performance.
• Use Krylov subspace solvers to solve the interface problem, with convergence monitored via iteration counts and far-field error.

Experimental results

Research questions

• RQ1 Can quasi-optimal transmission operators based on DtN approximations significantly reduce the number of iterations in DDM for Helmholtz transmission problems?
• RQ2 How does the convergence of DDM with quasi-optimal transmission operators scale with increasing frequency and material contrast?
• RQ3 Does the DDM framework remain robust and efficient when applied to partially coated domains with perfect electric conductor (PEC) boundaries?
• RQ4 How does the number of subdomains affect the iteration count in the proposed DDM with quasi-optimal transmission conditions?
• RQ5 Can the proposed method outperform standard CFIESK formulations in terms of iteration count and accuracy for high-contrast Helmholtz problems?

Key findings

• For a circular domain with ε₁ = 16 and ω = 64, the DDM with quasi-optimal transmission operators $Z_j$ required only 23 iterations, compared to 2271 for the CFIESK formulation.
• In the case of a partially coated circular domain, the DDM with $Z_j$ reduced iterations from 2271 (CFIESK) to 23 at ω = 32, with far-field errors below 8.5×10⁻³.
• When the domain was subdivided into four quarter-circle subdomains, the DDM with $Z_j$ required 173 iterations at ω = 32, showing minimal growth with frequency.
• The DDM with $Z_j^{PS}$ required 256 iterations at ω = 32 for two subdomains, while $Z_j^a$ required 529, indicating that the choice of transmission operator significantly affects convergence speed.
• For the L-shaped domain divided into three subdomains, the DDM with $Z_j$ required 447 iterations at ω = 32, demonstrating scalability with subdomain count.
• The far-field errors across all formulations remained below 8.5×10⁻³, indicating that the accuracy of the DDM is comparable to or better than the CFIESK method despite fewer iterations.