Direct-Indirect Mixed BM Boundary Integral Eq

• Direct-Indirect Mixed BM Equation is a formulation combining direct and indirect boundary representations to ensure well-posedness and eliminate spurious resonances.
• It leverages second-kind integral operators and mixed layer potentials, enabling efficient discretization and fast direct solvers for high-fidelity simulations.
• Applications span Helmholtz, elastodynamic, and electromagnetic problems, offering robust performance for complex transmission and scattering scenarios.

The direct-indirect mixed Burton-Miller (BM) boundary integral equation is a formulation for wave transmission, scattering, and resonance problems that combines direct and indirect boundary representations and employs the so-called Burton-Miller combination to eliminate spurious resonances. This approach is applied across the Helmholtz, elastodynamic, and electromagnetic contexts; it is especially effective for transmission across complex inclusions and in the presence of interior resonances. Recent research has established its injectivity, well-posedness, and significant numerical advantages—especially in conjunction with high-performance fast direct solvers—over conventional BM or direct-only formulations (Matsumoto et al., 14 May 2025, Matsumoto et al., 16 Dec 2025, Matsumoto et al., 24 Sep 2025).

1. Mathematical Formulation

Let $\Omega_1 \subset \mathbb{R}^d$ ($d=2,3$) be a bounded $C^2$ inclusion with boundary $\Gamma = \partial \Omega_1$ and exterior domain $$\Omega_0 = \mathbb{R}^d \setminus \overline{\Omega}_1$$. The Helmholtz transmission problem requires

$$\Delta u_j + k_j^2 u_j = 0 \quad \text{in} \;\Omega_j,\quad j=0,1, \ u_0^+ + u^{inc} = u_1^-, \quad \epsilon_0^{-1}\partial_\nu(u_0^+ + u^{inc}) = \epsilon_1^{-1}\partial_\nu u_1^- \quad \text{on } \Gamma,$$

with Sommerfeld radiation in $\Omega_0$. The direct-indirect mixed BM ansatz expands the solution using both single- and double-layer boundary potentials on $\Gamma$, and introduces auxiliary layer densities to facilitate second-kind integral operator properties. For a circular inclusion, the system for densities $\varphi,\psi$ reads (Matsumoto et al., 14 May 2025): $\begin{pmatrix} \frac12 I + D_{k_0} - i\eta\,S_{k_0} & - S_{k_0} \[6pt] - \alpha N_{k_1} + i\eta\left( \frac12 I - D_{k_1}' \right) & \frac12 I - D_{k_1}' + i\eta N_{k_1} \end{pmatrix} \begin{pmatrix} \varphi \ \psi \end{pmatrix} = \begin{pmatrix} g_D \ g_N \end{pmatrix},$ where $S_k, D_k, D_k', N_k$ are the single-layer, double-layer, adjoint double-layer, and hypersingular boundary operators, and $\eta\neq 0$ couples the combined field.

For general $C^2$ domains, the formulation enlarges the system to three unknowns by separating the indirect density for the interior and direct traces for the exterior (Matsumoto et al., 16 Dec 2025). The operator mapping is in Hölder spaces $$C^{1,\alpha} \times C^{0,\alpha} \times C^{0,\alpha}$$ and preserves key mapping and compactness properties.

2. Well-Posedness and Injectivity

The core analytical property is that the direct-indirect mixed BM operator is Fredholm of index zero and, for non-resonant configurations (i.e., $\eta \neq 0$, avoidance of Bessel and Hankel function derivative zeros for circular inclusions), is injective and hence invertible (Matsumoto et al., 14 May 2025). The operator’s block structure allows a direct proof by Fourier-mode decomposition for circular cases: the system decouples across angular frequencies, with each mode reducing to a $2\times 2$ or $3\times3$ linear system whose determinant is nonzero off resonance.

For curves of class $C^2$, injectivity and invertibility are established via mapping properties in Hölder spaces, standard jump relations for layer potentials, and block-triangular Fredholm arguments (Matsumoto et al., 16 Dec 2025). The mixed system’s injectivity implies no spurious “fictitious” eigenvalues—uniqueness holds for all real frequencies.

This property aligns the injectivity condition of the mixed BM operator with that of the ordinary BM operator for the same transmission setting—critical for applications in scattering by transmissive or resonant inclusions (Matsumoto et al., 14 May 2025).

3. Numerical Discretization

Discretization of the mixed BM integral equations is generally performed using Nyström or mixed Galerkin methods adapted to the operator’s structure:

• For elastodynamic and Helmholtz transmission, piecewise-linear bases are typically used for field traces, and piecewise-constant bases for normal derivatives (tractions), yielding square discretization matrices and optimal convergence (Matsumoto et al., 24 Sep 2025, Bao et al., 2016).
• For circular and smooth geometries, Fourier-based spectral discretizations allow decoupling into diagonal blocks (Matsumoto et al., 14 May 2025, Faria et al., 2023).
• The boundary integral operators’ singularities are handled using regularized quadrature or analytic power-log expansions for weakly singular and hypersingular kernels (Bao et al., 2016).

The redundancy of additional unknowns is offset by the resulting block-sparsity: the three-variable system admits efficient block LU factorizations with leading-order flop counts strictly lower than the ordinary BM formulation (e.g., $(14/3)N^3$ vs. $(16/3)N^3$ in 3-block vs. 2-block systems, respectively) (Matsumoto et al., 16 Dec 2025).

4. Fast Direct Solvers

A central numerical advantage of the direct-indirect mixed BM system is its compatibility with fast direct solvers leveraging low-rank off-diagonal compressions:

• Proxy method: The coefficient matrix is blocked according to the physical grouping of discretization points; off-diagonal blocks (well-separated boundary segments) are compressed using proxy point skeletonization and interpolative decomposition, leading to $O(N)$ complexity in 2D (Matsumoto et al., 24 Sep 2025, Matsumoto et al., 16 Dec 2025).
• Block LU structure: The three-unknown system allows local elimination of block rows/columns at each leaf in the hierarchical tree, reducing the dense system to a smaller skeleton system, which is efficiently solved and back-substituted (Matsumoto et al., 16 Dec 2025).
• Performance: Numerical experiments for both Helmholtz and elastodynamic transmission show 20–40% faster direct solve times for the mixed BM system relative to the ordinary BM or PMCHWT formulations, particularly in multi-RHS scenarios and at high degrees of freedom (up to $10^7$ unknowns) (Matsumoto et al., 24 Sep 2025, Matsumoto et al., 16 Dec 2025).
• Iterative methods: Analytical preconditioning is less effective for mixed discretizations due to the lack of a Calderón-like block structure; hence, direct solvers are strongly preferred in these contexts.

5. Application Domains

The direct-indirect mixed BM equation is applicable across multiple wave physics domains:

• Helmholtz transmission (scalar acoustics, optics): Provides uniquely solvable boundary integral representations even in the presence of inclusions or layered structures at all frequencies (Matsumoto et al., 16 Dec 2025, Toshimitsu et al., 2023).
• Elastodynamic transmission (elastic wave scattering by inclusions): Efficiently handles transmission conditions and complex geometries, leveraging mixed Galerkin discretizations and fast solvers (Matsumoto et al., 24 Sep 2025, Bao et al., 2016).
• Electromagnetic scattering: Combined field-only approaches utilizing a direct-indirect BM framework guarantee well-posedness for spherical and smooth PEC obstacles and possess favorable clustering of eigenvalues as well as explicit modal invertibility (Faria et al., 2023).
• Hybrid and half‐space boundary element methods: In hybrid representations coupling Sommerfeld integrals and layer potentials, the BM combination ensures resonance-free solutions, extends naturally to locally perturbed boundaries, and supports cavity scattering via virtual boundaries (Toshimitsu et al., 2023).

6. Modal and Spectral Analysis

On canonical geometric settings (circle, sphere), the modal structure allows explicit diagonalization:

• The mixed BM operator reduces to mode-by-mode block systems, each with characteristic determinant conditions.
• Avoidance of spurious resonances is ensured by choosing the coupling parameter $\eta$ away from “exceptional” values, and by keeping problem wavenumbers away from internal Dirichlet or Neumann eigenfrequencies (Matsumoto et al., 14 May 2025).
• For field-only BM on the sphere, the eigenvalues of the operator cluster at one, granting the discrete system the beneficial conditioning associated with second-kind integral equations (Faria et al., 2023).

7. Numerical Eigenvalue Problems and Broader Impact

The explicit structure, well-posedness, and multiple-right-hand-side efficiency of the mixed BM system facilitate advanced computational tasks:

• Nonlinear eigenvalue searches (e.g., Sakurai–Sugiura quadrature): The algorithmic reduction and explicit solution formulae enable high-accuracy, high-throughput contour integral-based root finding for resonance problems (Matsumoto et al., 16 Dec 2025).
• Large-scale and high-frequency scattering: The method scales efficiently for dense, high-frequency transmission problems in both acoustic and elastic regimes (Matsumoto et al., 24 Sep 2025, Cao et al., 2013).
• Broad flexibility: The approach is robust to boundary shape, frequency, material contrast, and allows incorporation of hybrid/indirect representations for open, layered, or multi-domain problems (Toshimitsu et al., 2023).

In summary, the direct-indirect mixed Burton-Miller boundary integral equation paradigm achieves provable well-posedness, resonance-free solvability, and state-of-the-art computational efficiency for boundary transmission problems, and has become foundational in advanced boundary element methods for acoustic, elastic, and electromagnetic scattering (Matsumoto et al., 16 Dec 2025, Matsumoto et al., 24 Sep 2025, Matsumoto et al., 14 May 2025).