Numerical simulations with rigid particles, drops or vesicles constitute some examples that involve 3D objects with spherical topology. When the numerical method is based on boundary integral equations, the error in using a regular quadrature rule to approximate the layer potentials that appear in the formulation will increase rapidly as the evaluation point approaches the surface and the integrand becomes sharply peaked. To determine when the accuracy becomes insufficient, and a more costly special quadrature method should be used, error estimates are needed. In this paper we present quadrature error estimates for layer potentials evaluated near surfaces of genus 0, parametrized using a polar and an azimuthal angle, discretized by a combination of the Gauss-Legendre and the trapezoidal quadrature rules. The error estimates involve no unknown coefficients, but complex valued roots of a specified distance function. The evaluation of the error estimates in general requires a one dimensional local root-finding procedure, but for specific geometries we obtain analytical results. Based on these explicit solutions, we derive simplified error estimates for layer potentials evaluated near spheres; these simple formulas depend only on the distance from the surface, the radius of the sphere and the number of discretization points. The usefulness of these error estimates is illustrated with numerical examples.

翻译：涉及刚性粒子、液滴或囊泡的数值模拟是典型的三维球形拓扑物体计算问题。当采用边界积分方程数值方法时，用常规求积法则逼近公式中的层势，会因求积误差随评价点趋近曲面而急剧增大（此时被积函数具有尖锐峰值）。为判定常规求积精度不足需改用更昂贵的特殊求积方法，需建立误差估计准则。本文针对采用极角与方位角参数化的零亏格曲面，结合Gauss-Legendre求积与梯形求积法则离散的层势近曲面评价问题，提出求积误差估计方法。该误差估计不含未知系数，但涉及特定距离函数的复值根。一般情况下需通过一维局部求根程序进行误差估计，但对特定几何构型可获解析结果。基于这些显式解，我们推导出球面附近层势评价的简化误差估计公式——该简洁表达式仅依赖于距曲面距离、球体半径与离散点数。通过数值算例验证了该误差估计的实用性。