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. Then, to regularize the integral, we multiply the Green’s function by a radial function with length parameter \(\delta \) chosen so that the product is smooth. 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, altered to account for the solvability condition, 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.

我们提出了一种简单而准确的方法来计算伴随双层势，该方法用于解决三维拉普拉斯方程的诺伊曼边值问题。曲线坐标的扩展导致我们修改伴随双层的表达式，以便在计算表面上的积分时减少奇异性。然后，为了使积分正则化，我们将格林函数乘以径向函数，并选择长度参数\(\delta \)以使乘积平滑。我们证明自然正则化具有误差\(O(\delta ^3)\)，并且简单的修改将误差改善为\(O(\delta ^5)\)。积分以数值方式计算，无需特殊坐标。我们使用这种伴随双层的处理来求解内部诺依曼问题的经典积分方程，并进行修改以考虑可解性条件，并评估边界上的解。选择\(\delta = ch^{4/5}\) ，我们在示例中发现大约\(O(h^4)\)收敛，其中h是背景网格中的间距。