Parallel One-Level Overlapping Schwarz Method

• Parallel one-level overlapping Schwarz method is a domain decomposition technique that divides the computational domain into overlapping subdomains and uses a partition of unity for assembling global solutions.
• It employs local PDE solvers with specific transmission conditions such as PML or impedance methods to achieve robust performance in high-frequency, indefinite, and wave propagation problems.
• The method scales effectively in complex geometries and is readily integrated with Krylov subspace solvers, ensuring practical convergence and reduced iteration counts.

The parallel one-level overlapping Schwarz method is a domain decomposition technique for solving partial differential equations (PDEs) by decomposing the computational domain into overlapping subdomains and iteratively solving local problems in parallel, combining local solutions to form global iterates. Its performance and theoretical properties have been rigorously analyzed for elliptic, parabolic, and time-harmonic equations, particularly in high-frequency and indefinite regimes where classical preconditioning techniques fail or scale poorly. Practical, scalable implementations have evolved, including variants with impedance or perfectly matched layer (PML) transmission conditions, especially tailored for the Helmholtz equation and related high-frequency wave problems.

1. Mathematical Framework

Let the computational domain $\Omega \subset \mathbb{R}^d$ be covered by $N$ overlapping Lipschitz subdomains $\{\Omega_j\}_{j=1}^N$. A smooth partition of unity $\{\chi_j\}$ is used such that $\sum_{j=1}^N \chi_j(x) \equiv 1$ on $\Omega_{int} \subset \Omega$. On each subdomain, a local PDE (possibly equipped with PML or impedance boundary treatments) is solved with Dirichlet or absorbing transmission conditions, and local corrections are restricted to the respective subdomain.

For the (scaled) Helmholtz equation with large wavenumber $k$ and variable wave speed $c(x)$,

$$-k^{-2}\Delta u - c^{-2}(x)u = f(x),\quad x\in \Omega,$$

truncated by a PML and discretized via finite elements, the Schwarz iteration proceeds as follows. At iteration $n$, for each $j$ in parallel:

1. Compute the local residual $r_j = (f - A u^n)|_{\Omega_j}$.
2. Solve the local problem $A_j c_j^n = r_j$ on the PML-augmented $\Omega_j$.
3. Update the global iterate: $u^{n+1} = u^n + \sum_{j=1}^N \chi_j c_j^n.$ The assembly and prolongation of local corrections employ the partition functions to ensure stability and consistency in the overlap regions (Galkowski et al., 2024).

2. Convergence Properties and Semiclassical Theory

Semiclassical analysis reveals that after a number of iterations bounded by the maximal number of subdomains intersected by a geometric-optic ray, the error is both smooth and arbitrarily small (in a negative power of $k$) in $k$-weighted Sobolev norms: $$\|u - u_*^{N_0}\|_{H_k^s(\Omega)} \le C k^{-M} \|u - u_*^0\|_{H_k^1(\Omega)}.$$ Here, $N_0$ is at most the number of subdomains a ray traverses (with multiplicity), $C$ is independent of $k$ and $f$, and $s,M$ are user-defined. In standard geometries (strip, checkerboard), $N_0$ matches subdomain counts or checker dimensions; for instance, for a $2 \times 2$ checkerboard, $N_0 = N_1 + N_2 - 1$ (Galkowski et al., 2024).

The theory is robust for fixed overlap and PML width (independent of $k$), but numerically, strong convergence is retained even as both decrease with $k$ (Xie et al., 31 Jan 2026).

3. Scalability, Transmission Conditions, and Absorbing Layers

Classical Dirichlet transmission in 1D or chain geometries leads to a contraction after approximately $N/2$ steps, with the number of required global Schwarz iterations scaling linearly with $N$—demonstrating a lack of weak scalability (Ciaramella et al., 2019). Generalizing to higher dimensions and more complex geometries, the contraction is achieved after a number of iterations equal to the graph-diameter of the overlap connectivity (“layer count”), and convergence degrades if the diameter increases with $N$ (Ciaramella et al., 2019).

Introducing absorbing (impedance) or PML transmission conditions achieves almost uniform contraction factors, with the limiting spectrum of the Schwarz iteration operator characterized by block Toeplitz-matrix theory; the spectral radius is strictly less than one given sufficient absorption or sufficiently strong overlapping/PML layers, ensuring scalability even as the number of subdomains increases (Bootland et al., 2020, Gong et al., 2021, Galkowski et al., 2024).

The use of PMLs as subdomain boundary transmission is particularly effective for wave-type problems, with robust performance for PML and overlap widths scaling as $O(k^{-1} \log k)$—enabling weak to strong scaling up to $O(k^d)$ subdomains for fixed per-subdomain workload (Xie et al., 31 Jan 2026, Galkowski et al., 2024).

4. Discrete and Operator Formulations

In discrete algebraic form, the parallel one-level additive Schwarz iteration corresponds to a Richardson iteration preconditioned by the restricted additive Schwarz (RAS) preconditioner: $$u^{n+1} = u^n + B^{-1}(f - A u^n),\quad B^{-1} = \sum_{j=1}^N \widetilde R_j^T A_j^{-1} R_j,$$ where $R_j$ restricts to $\Omega_j$, $A_j$ is the local stiffness matrix, and $\widetilde R_j^T$ applies the partition-weighted prolongation. The fixed-point iteration operator $E_{add} = I - B^{-1}A$ governs the convergence, and the algebraic structure is crucial in analysis and efficient parallel implementation (Galkowski et al., 2024, Xie et al., 31 Jan 2026).

The method is naturally compatible with Krylov subspace solvers; using Schwarz as a preconditioner for GMRES yields further acceleration and robustness as $N$ or $k$ increases (Bootland et al., 2020, May et al., 2019).

5. Numerical Evidence and Practical Parameter Choices

Empirical studies confirm theoretical predictions. For the Helmholtz equation with high $k$ and minimal overlap or PML (as thin as one wavelength), the number of additive Schwarz iterations remains bounded or grows mildly with $k$ or $N$. Detailed data for checkerboard decompositions and varying overlap/PML sequences show essentially no sensitivity to overlap width for prescribed accuracy:

• Strip decompositions: Iteration count $\lesssim N$.
• Checkerboard: Iteration count $\lesssim N_1 + N_2 - 1$.
• Minimal overlap ($\delta = h$ or $2h$) yields the same convergence as wider overlaps, with dramatically reduced communication.
• Convergence is even faster at higher $k$, with the per-iteration convergence factor $\rho = O(k^{-\alpha}),~\alpha \approx 0.2$–$0.3$ (Galkowski et al., 2024).

Recommendation summary:

• Set PML width to one free-space wavelength ($\kappa \approx 2\pi/k$).
• Use polynomial-type scaling in the PML profile for low reflection (e.g., cubic with coefficient proportional to $k$).
• Restrict overlap to minimal layers (one or two elements).
• Choose subdomain aspect ratios to optimize parallel efficiency and iteration bound.

6. Domain Generalization and Further Extensions

The method generalizes across PDE classes (Laplace, convection-diffusion, parabolic, Helmholtz, Maxwell, and constrained QP systems) and to higher dimensions. For elliptic and parabolic problems, overlapping Schwarz with Dirichlet transmission is always convergent, and condition numbers are bounded by $O(1 + H/\delta)$, where $H$ is the subdomain size and $\delta$ the overlap (Oh et al., 2024, Aiton et al., 2019, Tran, 2010). For indefinite or highly oscillatory problems, well-chosen Robin, impedance, or PML boundary treatments are essential for scalability and robustness.

Ongoing work includes two-level extensions (with global coarse corrections), multi-physics coupling, general communication patterns (e.g., in graphs or on surfaces), and high-performance parallel implementation on massively distributed-memory architectures (May et al., 2019, Xie et al., 31 Jan 2026).

7. Summary Table: Algorithmic and Convergence Features

Feature	Description / Guidance	Ref.
Transmission condition	Dirichlet / Impedance / PML	(Galkowski et al., 2024, Bootland et al., 2020)
Overlap width	$\delta \sim h$ or $O(k^{-1} \log k)$	(Galkowski et al., 2024, Xie et al., 31 Jan 2026)
PML width	$\kappa \approx 2\pi/k$	(Galkowski et al., 2024)
Iteration count	$\lesssim N$, nearly $k$-independent	(Galkowski et al., 2024, Xie et al., 31 Jan 2026)
Preconditioned iteration count	Near constant for Krylov methods	(Bootland et al., 2020)
Scalability	Weak: O(1) iters if overlap-graph diameter $=O(1)$; strong for PML/impedance	(Ciaramella et al., 2019, Xie et al., 31 Jan 2026)

• Galkowski et al., "Schwarz methods with PMLs for Helmholtz problems: fast convergence at high frequency" (Galkowski et al., 2024)
• Xie et al., "Massively parallel Schwarz methods for the high frequency Helmholtz equation" (Xie et al., 31 Jan 2026)
• Lafontaine et al., "Convergence of parallel overlapping domain decomposition methods for the Helmholtz equation" (Gong et al., 2021)
• Gander et al., "Analysis of parallel Schwarz algorithms for time-harmonic problems using block Toeplitz matrices" (Bootland et al., 2020)