Efficient LU factorization exploiting direct-indirect Burton-Miller equation for Helmholtz transmission problems

TL;DR

The paper tackles Helmholtz transmission problems by introducing a direct-indirect Burton–Miller boundary integral equation, which adds one auxiliary unknown but reorganizes the operator structure to be sparsely block-aligned. This enables efficient LU-factorization-based direct solvers that outperform the standard Burton–Miller formulation by roughly 40% in the tested cases, and extends to a fast direct solver using proxy-based low-rank compression under weak admissibility. The authors prove well-posedness in Hölder spaces, extend the method to general right-hand sides for nonlinear eigenvalue analyses (via Sakurai–Sugiura), and demonstrate accurate eigenvalue recovery alongside substantial computational savings. While memory requirements increase due to extra DOFs, the approach remains competitive, with potential for HPC-scale extensions and applicability to other PDEs such as Maxwell or elastic systems.

Abstract

This paper proposes a direct-indirect mixed Burton-Miller boundary integral equation for solving Helmholtz scattering problems with transmissive scatterers. The proposed formulation has three unknowns, one more than the number of unknowns for the ordinary formulation. However, we can construct efficient numerical solvers based on LU factorization by exploiting the sparse alignment of the boundary integral operators of the proposed formulation. Numerical examples demonstrate that the direct solver based on the proposed formulation is approximately 40% faster than the ordinary formulation when the LU-factorization-based solver is used. In addition, the proposed formulation is applied to a fast direct solver employing LU factorization in its algorithm. In the application to the fast direct solver, the proxy method with a weak admissibility low-rank approximation is developed. The speedup achieved using the proposed formulation is also shown to be effective in finding nonlinear eigenvalues, which are related to the uniqueness of the solution, in boundary value problems. Furthermore, the well-posedness of the proposed boundary integral equation is established for scatterers with boundaries of class $C^2$, using the mapping property of boundary integral operators in Hölder space.

Figures (6)

• Figure 1: Diagram of the Helmholtz transmission problem
• Figure 2: Local virtual boundary (proxy boundary) in the proxy method
• Figure 3: Relative 2-norm errors of numerical solution of each formulation with respect to analytical solution obtained using direct solver based on LU factorization; the figure presents two formulations for each pair of frequency ($\omega$) and internal material parameter ($\varepsilon_1$), namely the discretized direct-indirect mixed Burton--Miller equation ("Mixed") and the discretized ordinary Burton--Miller equation ("Ordinary")
• Figure 4: Computational time versus $N$ obtained using LU factorization for solving discretized direct-indirect mixed Burton--Miller equation ("Mixed") and discretized ordinary Burton--Miller equation ("Ordinary") in each pair of frequency $\omega$ and material parameter inside $\varepsilon_1$; only the mixed equation benefits from Theorem \ref{['thm:LU']}
• Figure 5: Relative 2-norm errors of numerical solution of each formulation with respect to analytical solution obtained using fast direct solver; the figure presents two formulations for each pair of frequency ($\omega$) and internal material parameter ($\varepsilon_1$), namely the discretized direct-indirect mixed Burton--Miller equation ("Mixed") and the discretized ordinary Burton--Miller equation ("Ordinary")

Theorems & Definitions (20)

• Remark 2.1
• Theorem 2.2: Colton and Kress colton2013inverse, Theorem 3.4, and Kress kress2014linear, Theorems 7.5 and 7.6
• Theorem 2.3: Colton and Kress colton1983integral, Theorem 2.31, and Colton and Kress colton2013inverse, § 3.1
• Theorem 2.4: Gilbarg and Trudinger gilbarg1998elliptic, Lemma 6.36
• Lemma 2.5
• proof
• Theorem 2.6: Kress kress2014linear, Theorem 3.4
• Lemma 3.1
• proof
• Lemma 3.2