Asymptotic-DLP

Asymptotic-DLP. This repository contains the codes used in the paper: Asymptotic approximations for the close evaluation of double-layer potentials. When using boundary integral equation methods to solve a boundary value problem, the evaluation of the solution near the boundary is challenging to compute because the layer potentials that represent the solution are nearly singular integrals. To address this close evaluation problem, we develop a new numerical method by applying an asymptotic analysis of these nearly singular integrals and obtaining an asymptotic approximation. We derive the asymptotic approximation for the case of the double-layer potential in two and three dimensions, representing the solution of the interior Dirichlet problem for Laplace’s equation. By doing so, we obtain an asymptotic approximation given by the Dirichlet data at the boundary point nearest to the interior evaluation point plus a nonlocal correction. We present the numerical methods using this asymptotic approximation, and we demonstrate the efficiency and accuracy of these methods and the asymptotic approximation through several examples. These examples show that the numerical method based on the asymptotic approximation accurately approximates the close evaluation of the double-layer potential while requiring only modest computational resources.

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