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Understanding and Resolving Singularities in 3D Dirichlet Boundary Problems

David Levin

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.

Understanding and Resolving Singularities in 3D Dirichlet Boundary Problems

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.
Paper Structure (21 sections, 2 theorems, 48 equations, 5 figures, 1 table)

This paper contains 21 sections, 2 theorems, 48 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries - Green's identity and Green's function
  3. Green's Function for the Half-Space $x_3\ge 0$
  4. The unit cube
  5. The cylinder
  6. The finite cylinder
  7. Reviewing the Corrected Collocation Method in 2D
  8. Challenges in Solving the Unit Cube Problem
  9. The S--R method
  10. Singular–Regular Decomposition of the Green’s Function
  11. The Singular-Regular (S--R) approach
  12. The approximation procedure
  13. Theoretical and implementation details for the unit cube
  14. Computing $\tilde{H}_R$ on $\partial\Omega$
  15. Choosing the collocation points
  16. ...and 6 more sections

Key Result

Corollary 3.4

In view of Assumption Rassumption, the solution admits the decomposition eq:SR where

Figures (5)

  • Figure 1: The unit cube; The source point in blue plus 26 triple reflections in red.
  • Figure 2: The figure shows the 150 interpolation points on the cube's faces, colored by their sampled values, along with the corresponding source points (marked by circles) used in the interpolation basis.
  • Figure 3:
  • Figure 4: The computed approximation $P_{150}$ of the regular component of the solution.
  • Figure 5: The figure displays the computed approximation in the vicinity of the cube corner. It clearly captures the singular transition from the value $1$ on the upper face to the value $0$ on the adjacent faces.

Theorems & Definitions (5)

  • Definition 3.1
  • Remark 3.2
  • Corollary 3.4
  • Proposition 3.7
  • proof