Understanding and Resolving Singularities in 3D Dirichlet Boundary Problems

TL;DR

This paper presents a novel two-phase approximation method for effectively resolving singularities in 3D harmonic Dirichlet boundary problems, combining Green's function decomposition with high-order quadrature and harmonic basis collocation.

Contribution

It introduces a new two-phase approach that decomposes the solution into singular and regular parts, improving accuracy in 3D Dirichlet problems.

Findings

Successfully resolves singularities in 3D harmonic Dirichlet problems

Employs high-order quadrature for accurate singularity handling

Uses harmonic basis collocation for the regular part

Abstract

We introduce a two-phase approximation method designed to resolve singularities in three-dimensional harmonic Dirichlet problems. The approach utilizes the classical Green's function representation, decomposing the function into its singular and regular components. The singular phase employs Green's formula with the singular part, for which we show that it induces the necessary singularities in the solution. The regular phase then introduces a smooth correction to recover the remaining regular part of the solution. The construction employs high-order quadrature rules in the first phase, followed by collocation with a suitable harmonic basis in the second.