A robust solver for elliptic PDEs in 3D complex geometries

by   Matthew J. Morse, et al.

We develop a boundary integral equation solver for elliptic partial differential equations on complex 3D geometries. Our method is high-order accurate with optimal O(N) complexity and robustly handles complex geometries. A key component is our singular and near-singular layer potential evaluation scheme, hedgehog : a simple extrapolation of the solution along a line to the boundary. We present a series of geometry-processing algorithms required for hedgehog to run efficiently with accuracy guarantees on arbitrary geometries and an adaptive upsampling scheme based on a iteration-free heuristic for quadrature error that incorporates surface curvature. We validate the accuracy and performance with a series of numerical tests and compare our approach to a competing local evaluation method.


page 22

page 27

page 42


Solution of Stokes flow in complex nonsmooth 2D geometries via a linear-scaling high-order adaptive integral equation scheme

We present a fast, high-order accurate and adaptive boundary integral sc...

A Fast Integral Equation Method for the Two-Dimensional Navier-Stokes Equations

The integral equation approach to partial differential equations (PDEs) ...

A reduced order Schwarz method for nonlinear multiscale elliptic equations based on two-layer neural networks

Neural networks are powerful tools for approximating high dimensional da...

Analytical computation of boundary integrals for the Helmholtz equation in three dimensions

A key issue in the solution of partial differential equations via integr...

Quadrature by fundamental solutions: kernel-independent layer potential evaluation for large collections of simple objects

Well-conditioned boundary integral methods for the solution of elliptic ...

Sparse recovery of elliptic solvers from matrix-vector products

In this work, we show that solvers of elliptic boundary value problems i...