A robust solver for elliptic PDEs in 3D complex geometries

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.