The adjoint double layer potential on smooth surfaces in \(\mathbb {R}^3\) and the Neumann problem

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 \(\delta \) $\delta$ chosen so that the product is smooth. We show that a natural regularization has error \(O(\delta ^3)\) $O({\delta }^3)$ , and a simple modification improves the error to \(O(\delta ^5)\) $O({\delta }^5)$ . 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 \(\delta = ch^{4/5}\) $\delta =c{h}^{4/5}$ , we find about \(O(h^4)\) $O(h^4)$ convergence in our examples, where h is the spacing in a background grid.