A robust solver for elliptic PDEs in 3D complex geometries

• Published in: Journal of computational physics Vol. 442; p. 110511
• Main Authors: Morse, Matthew J., Rahimian, Abtin, Zorin, Denis
• Format: Journal Article
• Language: English
• Published: Cambridge Elsevier Inc 01.10.2021; Elsevier Science Ltd
• Subjects: Algorithms; Boundary integral method; Computational physics; Differential geometry; Elliptic differential equations; Elliptic functions; Geometric accuracy; Integral equations; Iterative methods; Materials handling; Partial differential equations; Quadratures; Robustness (mathematics); Solvers
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2021.110511

•A fast accurate singular/near-singular quadrature scheme on complex 3D geometries is presented.•Robust geometry processing algorithms are detailed to support quadrature.•The proposed method is tested on non-trivial geometries and topologies and compared against a competing approach. We develop a boundary integral equation solver for elliptic partial differential equations on complex 3D geometries. Our method is efficient, high-order accurate 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. We validate the accuracy and performance with a series of numerical tests and compare our approach to a competing local evaluation method.