The implicit boundary integral method (IBIM) provides a framework to construct quadrature rules on regular lattices for integrals over irregular domain boundaries. This work provides a systematic error analysis for IBIMs on uniform Cartesian grids for boundaries with different degree of regularities. We first show that the quadrature error gains an addition order of $\frac{d-1}{2}$ from the curvature for a strongly convex smooth boundary due to the ``randomness'' in the signed distances. This gain is discounted for degenerated convex surfaces. We then extend the error estimate to general boundaries under some special circumstances, including how quadrature error depends on the boundary's local geometry relative to the underlying grid. Bounds on the variance of the quadrature error under random shifts and rotations of the lattices are also derived.

翻译：隐式边界积分方法（IBIM）提供了在规则网格上为不规则区域边界积分构造求积规则的框架。本文针对具有不同正则程度的边界，对均匀笛卡尔网格上的IBIM进行了系统的误差分析。我们首先证明，对于强凸光滑边界，由于符号距离的“随机性”，求积误差会从曲率中获得额外的$\frac{d-1}{2}$阶精度提升。对于退化凸曲面，这一提升会减弱。随后，我们将误差估计扩展到一般边界在特定情况下的分析，包括求积误差如何依赖于边界相对于底层网格的局部几何形状。本文还推导了网格随机平移和旋转下求积误差方差的界。