[Paper Review] Domain Decomposition with local impedance conditions for the Helmholtz equation

Ivan G. Graham, Euan A. Spence+1 more|arXiv (Cornell University)|Jun 10, 2018

Advanced Numerical Methods in Computational MathematicsEngineering16 citations

TL;DR

This paper proposes a one-level additive Schwarz preconditioner for the Helmholtz equation using local impedance conditions on overlapping subdomains, combined with partition-of-unity prolongation. It achieves k-robust GMRES convergence for increasing wavenumbers k, supported by theoretical bounds on the preconditioned matrix norm and field of values in the energy inner product.

ABSTRACT

We consider one-level additive Schwarz preconditioners for the equation (with increasing wavenumber $k$), discretized using fixed-order nodal conforming finite elements on a family of simplicial fine meshes with diameter $h$, chosen to maintain accuracy as $k$ increases. The preconditioners combine independent local solves (with impedance boundary conditions) on overlapping subdomains of diameter $H$ and overlap $\delta$, and prolongation/restriction operators defined using a partition of unity, this formulation was previously proposed in [J.H. Kimn and M. Sarkis, Comp. Meth. Appl. Mech. Engrg. 196, 1507-1514, 2007]. In numerical experiments (with $\delta \sim H$) we observe robust (i.e. $k-$independent) GMRES convergence as $k$ increases, both with $H$ fixed, and with $H$ decreasing moderately as $k$ increases. This provides a highly-parallel, $k-$robust one-level domain-decomposition method. We provide supporting theory for this observation by studying the preconditioner applied to a range of absorptive problems, $k^2\mapsto k^2+ \mathrm{i} \varepsilon$, with absorption parameter $\varepsilon$, including the Helmholtz case ($\varepsilon = 0$). Working in the energy inner product, we prove a robust upper bound on the norm of the preconditioned matrix, valid for all $\varepsilon, \delta$. Under additional conditions on $\varepsilon$ and $\delta$, we also prove a strictly-positive lower bound on the distance of the field of values of the preconditioned matrix from the origin. Using these results, combined with previous results of [M.J. Gander, I.G. Graham and E.A. Spence, Numer. Math. 131(3), 567-614, 2015] we obtain theoretical support for the observed robustness of the preconditioner for the pure problem with increasing wavenumber $k$.

Motivation & Objective

• To develop a domain decomposition preconditioner that maintains robust convergence for the Helmholtz equation as the wavenumber k increases.
• To analyze the behavior of the preconditioner in the high-frequency regime using absorptive formulations with complex shift ε.
• To provide theoretical justification for the observed k-robustness in numerical experiments through bounds on the preconditioned matrix.
• To establish conditions under which the preconditioned matrix remains well-conditioned and its field of values stays bounded away from the origin.

Proposed method

• The method employs overlapping subdomains of diameter H and overlap δ, with local solves using impedance boundary conditions on each subdomain.
• Prolongation and restriction operators are constructed using a partition of unity to ensure consistency and stability.
• The preconditioner is analyzed in the energy inner product for a family of absorptive Helmholtz problems with k² → k² + iε.
• Theoretical bounds are derived for the norm of the preconditioned matrix, valid for all ε > 0 and δ > 0.
• Additional conditions on ε and δ are introduced to prove a strictly positive lower bound on the distance of the field of values from the origin.
• The results are extended to the pure Helmholtz case (ε = 0) using prior theoretical work on field of values and GMRES convergence.

Experimental results

Research questions

• RQ1 Can a one-level additive Schwarz preconditioner with local impedance conditions achieve k-robust convergence for the Helmholtz equation?
• RQ2 What theoretical conditions on ε and δ ensure that the preconditioned matrix remains well-conditioned in the high wavenumber limit?
• RQ3 How do the energy norm bounds on the preconditioned matrix support the observed k-robustness in numerical experiments?
• RQ4 What is the role of the partition of unity in maintaining robustness across varying k and mesh refinement?

Key findings

• A robust upper bound on the norm of the preconditioned matrix is proven for all ε > 0 and δ > 0, independent of k.
• Under additional conditions on ε and δ, a strictly positive lower bound is established on the distance of the field of values from the origin, ensuring well-conditioning.
• Theoretical support is provided for the observed k-robust GMRES convergence in numerical experiments, even as k increases.
• The method remains effective both with fixed H and with H decreasing moderately as k increases, demonstrating practical robustness.