A Fast Direct Solver for Boundary Integral Equations Using Quadrature By Expansion
This work provides an efficient and accurate direct solver for boundary integral equations, benefiting computational scientists in electromagnetics, acoustics, and fluid dynamics.
The authors develop a hierarchical direct solver for boundary integral equations using Quadrature by Expansion (QBX), achieving state-of-the-art asymptotic scaling with automatic error control. Numerical experiments confirm theoretical error and cost predictions.
We construct and analyze a hierarchical direct solver for linear systems arising from the discretization of boundary integral equations using the Quadrature by Expansion (QBX) method. Our scheme builds on the existing theory of Hierarchical Semi-Separable (HSS) matrix operators that contain low-rank off-diagonal submatrices. We use proxy-based approximations of the far-field interactions and the Interpolative Decomposition (ID) to construct compressed HSS operators that are used as fast direct solvers for the original system. We describe a number of modifications to the standard HSS framework that enable compatibility with the QBX family of discretization methods. We establish an error model for the direct solver that is based on a multipole expansion of the QBX-mediated proxy interactions and standard estimates for the ID. Based on these theoretical results, we develop an automatic approach for setting scheme parameters based on user-provided error tolerances. The resulting solver seamlessly generalizes across two- and tree-dimensional problems and achieves state-of-the-art asymptotic scaling. We conclude with numerical experiments that support the theoretical expectations for the error and computational cost of the direct solver.