We present a simple yet accurate method to compute the adjoint double layer potential, which is used to solve the Neumann boundary value problem for Laplace's equation in three dimensions. An expansion in curvilinear coordinates leads us to modify the expression for the adjoint double layer so that the singularity is reduced when evaluating the integral on the surface. We then regularize the Green's function, with a radial parameter $\delta$. We show that a natural regularization has error $O(\delta^3)$, and a simple modification improves the error to $O(\delta^5)$. The integral is evaluated numerically without the need of special coordinates. We use this treatment of the adjoint double layer to solve the classical integral equation for the interior Neumann problem and evaluate the solution on the boundary. Choosing $\delta = ch^{4/5}$, we find about $O(h^4)$ convergence in our examples, where $h$ is the spacing in a background grid.

翻译：我们提出一种简单而精确的方法来计算伴随双层势，用于求解三维拉普拉斯方程诺伊曼边值问题。通过曲线坐标展开，我们修改了伴随双层的表达式，从而降低了曲面上的积分奇异性。随后，我们使用径向参数 $\delta$ 对格林函数进行正则化。我们证明，自然正则化的误差为 $O(\delta^3)$，而简单的修改可将误差改进至 $O(\delta^5)$。该积分无需特殊坐标即可进行数值计算。我们利用这种伴随双层处理方法求解经典的内诺伊曼问题积分方程，并在边界上评估解。选取 $\delta = ch^{4/5}$，我们在示例中观察到约 $O(h^4)$ 的收敛速度，其中 $h$ 为背景网格的间距。