FMM-LU: A fast direct solver for multiscale boundary integral equations in three dimensions

TL;DR

This paper introduces a fast direct solver for 3D boundary integral equations that achieves linear computational complexity for non-oscillatory problems, enabling efficient solutions for complex surface problems in physics.

Contribution

It extends recursive strong skeletonization to 3D boundary integral equations, providing a linear-time LU-like factorization compatible with adaptive octree discretizations.

Findings

Achieves O(n) complexity for applying the inverse.

Supports highly nonuniform discretizations.

Demonstrates effectiveness in acoustic scattering problems.

Abstract

We present a fast direct solver for boundary integral equations on complex surfaces in three dimensions using an extension of the recently introduced recursive strong skeletonization scheme. For problems that are not highly oscillatory, our algorithm computes an ${LU}$-like hierarchical factorization of the dense system matrix, permitting application of the inverse in $\mathcal O(n)$ time, where $n$ is the number of unknowns on the surface. The factorization itself also scales linearly with the system size, albeit with a somewhat larger constant. The scheme is built on a level-restricted adaptive octree data structure, and therefore it is compatible with highly nonuniform discretizations. Furthermore, the scheme is coupled with high-order accurate locally-corrected Nystr\"om quadrature methods to integrate the singular and weakly-singular Green's functions used in the integral representations. Our method has immediate applications to a variety of problems in computational physics. We concentrate here on studying its performance in acoustic scattering (governed by the Helmholtz equation) at low to moderate frequencies, and provide rigorous justification for compression of submatrices via proxy surfaces.