Convergence of parallel overlapping domain decomposition methods with impedance boundary conditions for time-harmonic Maxwell equations in heterogeneous media

Technical Summary: Convergence of Parallel Overlapping Domain Decomposition Methods for Time-Harmonic Maxwell Equations

Problem Statement
This paper addresses the numerical solution of the time-harmonic Maxwell equations in heterogeneous media, a problem central to applications such as radar, wireless communications, and medical imaging. At high frequencies, these problems present significant challenges: the need for fine meshes to resolve oscillations leads to large, non-Hermitian, and highly indefinite linear systems. While two-level domain decomposition (DD) methods have established convergence theories (often relying on coarse spaces that resolve wave propagation), there is a lack of rigorous convergence theory for one-level overlapping DD methods for Maxwell equations, particularly in heterogeneous settings. The authors note that standard field-of-values arguments used for two-level methods fail for the one-level methods considered here, as the field of values contains the origin.

Methodology
The authors analyze a parallel overlapping Schwarz method with impedance boundary conditions on subdomain interfaces. The methodology proceeds in two main stages: continuous analysis and discrete analysis.

1. Continuous Level Analysis:

   • Well-posedness: The paper establishes the well-posedness of the parallel Schwarz iteration in appropriate function spaces ($X_{imp}$ and $U$) for general Lipschitz subdomains. This relies on harmonic-analysis results regarding the regularity of Maxwell solutions. A key technical hurdle addressed is the regularity of the partition of unity functions and the resulting error terms, which requires specific assumptions (Assumption 2.11) regarding the geometry of strip decompositions.
   • Error Propagation: The error propagation operator is characterized using impedance-to-impedance maps. These maps describe how impedance data (tangential traces of the curl and the field itself) on one subdomain interface propagates to neighboring interfaces.
   • Strip Decompositions: For strip domain decompositions, the authors derive explicit convergence estimates. They show that the convergence of the $N$-th power of the error propagation operator depends on the norms of specific impedance-to-impedance maps (denoted $\rho$ and $\gamma$), which represent "left-to-right" and "left-to-left" (or "right-to-right") interactions.

2. Discrete Level Analysis:

   • Discretization: The continuous method is discretized using Nédélec finite elements. The discrete parallel Schwarz method is identified as a Restricted Additive Schwarz (RAS-imp) method preconditioning a Richardson iteration.
   • Discrete Error Propagation: The authors formulate a discrete error propagation operator in terms of discrete impedance-to-impedance maps.
   • Convergence Inheritance: Under the assumption that the discrete impedance-to-impedance maps converge to their continuous counterparts as the mesh size $h \to 0$, the paper proves that the discrete method inherits the convergence behavior of the continuous method.

Key Contributions

• PDE-Level Framework: Development of a framework for parallel overlapping Schwarz methods with impedance transmission conditions for variable-coefficient Maxwell equations. This includes proving well-posedness for general Lipschitz subdomains and deriving error equations expressed via impedance-to-impedance maps.
• Convergence Estimates for Strips: Derivation of sufficient conditions for convergence in strip decompositions. The analysis reduces the problem of bounding the full error propagation operator to bounding the norms of interface maps ($\rho$ and $\gamma$).
• Discrete Convergence Theory: Demonstration that the Nédélec discretization (RAS-imp) inherits the convergence properties of the continuous method, provided the discrete interface maps approximate the continuous ones.
• Numerical Validation: Extensive numerical experiments validating the theory. These include:
  • Verification of the power contractivity of the discrete error propagation operator.
  • Computation of the norms of impedance-to-impedance maps, confirming they are bounded and consistent with theoretical predictions.
  • Performance tests on both strip and checkerboard decompositions in homogeneous and heterogeneous media, showing that RAS-imp acts effectively as both a stationary iteration and a Krylov preconditioner (for GMRES).

Results

• Theoretical Bounds: The paper establishes that for strip decompositions, the error propagation operator $T^N$ becomes contractive if the impedance-to-impedance map norms are sufficiently small. Specifically, the norm of the $N$-th power of the operator is bounded by a function of $\rho$ and $\gamma$.
• Numerical Observations:
  • In strip decompositions, the residual history exhibits a "stepwise decay," corresponding to the power contractivity where the error norm drops significantly after approximately $N$ iterations (where $N$ is the number of subdomains).
  • The impedance-to-impedance map norms ($\rho$ and $\gamma$) were computed numerically. $\rho$ (representing transmission across the domain) was found to be well below 1, while $\gamma$ (representing reflection/interaction within the same direction) remained close to 1 but bounded.
  • The method demonstrated robustness with respect to the wavenumber $k$ (for fixed $N$) and scalability with respect to the number of subdomains.
  • For checkerboard decompositions (which are beyond the strict theoretical scope of the strip analysis), the method still exhibited power contractivity, though the threshold for contraction shifted to approximately $2\sqrt{N}$ iterations.
  • In heterogeneous media, convergence was generally slower than in homogeneous cases, attributed to internal reflections caused by spatially varying permittivity, but the method remained effective.

Significance and Claims
The paper claims to provide the first convergence theory for one-level overlapping domain decomposition methods for time-harmonic Maxwell equations at both the continuous and discrete levels. The authors emphasize that this work fills a gap left by existing two-level theories, which often rely on coarse spaces that obscure the propagation behavior crucial to one-level methods.

The authors are modest regarding the extent of their results compared to analogous work on the Helmholtz equation (cited as [20, 21, 27]). They explicitly note three limitations:

1. The well-posedness result for the continuous iteration requires a regularity assumption (Assumption 2.11) that is not yet proven for general Lipschitz subdomains in the Maxwell case, unlike the Helmholtz case.
2. The convergence of discrete impedance-to-impedance maps to continuous ones is assumed (Assumption 6.5) rather than proven, due to the reduced regularity of Maxwell solutions and the complexity of the necessary duality arguments.
3. The paper does not provide high-frequency bounds for the impedance-to-impedance maps at the continuous level, as the semiclassical defect measure theory required for such bounds is not yet established for time-harmonic Maxwell equations.

Despite these limitations, the paper asserts that the developed framework successfully characterizes the error propagation via interface maps and that the numerical experiments strongly support the theoretical predictions, validating RAS-imp as a robust solver for these challenging electromagnetic problems.