2016/12/03 by Abtin Rahimian, Rahimian, Abtin, Alex H. Barnett +3
Engineering · Physics and Astronomy · #Advanced Numerical Methods in Computational Mathematics #Electromagnetic Scattering and Analysis #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in engineering
paper · pdf · doi:10.48550/arxiv.1612.00977
openalex publication_date 2016/12/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce a quadrature scheme--QBKIX--for the high-order accurate\nevaluation of layer potentials associated with general elliptic PDEs near to\nand on the domain boundary. Relying solely on point evaluations of the\nunderlying kernel, our scheme is essentially PDE-independent; in particular, no\nanalytic expansion nor addition theorem is required. Moreover, it applies to\nboundary integrals with singular, weakly singular, and hypersingular kernels.\n Our work builds upon Quadrature by Expansion (QBX), which approximates the\npotential by an analytic expansion in the neighborhood of each expansion\ncenter. In contrast, we use a sum of fundamental solutions lying on a ring\nenclosing the neighborhood, and solve a small dense linear system for their\ncoefficients to match the potential on a smaller concentric ring.\n We test the new method with Laplace, Helmholtz, Yukawa, Stokes, and Navier\n(elastostatic) kernels in two dimensions (2D) using adaptive, panel-based\nboundary quadratures on smooth and corner domains. Advantages of the algorithm\ninclude its relative simplicity of implementation, immediate extension to new\nkernels, dimension-independence (allowing simple generalization to 3D), and\ncompatibility with fast algorithms such as the kernel-independent FMM.\n