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

TL;DR

This paper introduces a novel boundary integral equation formulation for Helmholtz transmission problems that enables faster LU-based solvers and improves efficiency in eigenvalue computations, with proven well-posedness for smooth boundaries.

Contribution

It develops a direct-indirect mixed Burton-Miller formulation with sparse operator structure, facilitating efficient LU factorization and fast direct solvers for Helmholtz transmission problems.

Findings

Approximate 40% speedup over ordinary formulation.

Effective application of LU-based direct solver with low-rank approximations.

Successful use in nonlinear eigenvalue computations.

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 H\"older space.