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.

刚性粒子、液滴或囊泡的数值模拟构成了涉及具有球形拓扑的 3D 对象的一些示例。当数值方法基于边界积分方程时，使用正则求积规则来近似公式中出现的层势的误差将随着评估点接近表面并且被积函数变得急剧峰值而迅速增加。为了确定何时精度变得不够，并且应该使用成本更高的特殊正交方法，需要进行误差估计。在本文中，我们提出了对类 0 表面附近评估的层势的正交误差估计，使用极角和方位角进行参数化，并通过高斯-勒让德和梯形求积规则的组合进行离散化。误差估计不涉及未知系数，而是指定距离函数的复值根。一般来说，误差估计的评估需要一维局部求根过程，但对于特定的几何形状，我们获得分析结果。基于这些显式解，我们得出了在球体附近评估的层势的简化误差估计；这些简单的公式仅取决于距表面的距离、球体的半径和离散点的数量。通过数值示例说明了这些误差估计的有用性。