Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Fast Fourier-Galerkin Method for Solving Boundary Integral Equations on Torus-Shaped Surfaces

Yiying Fang, Ying Jiang, Jiafeng Su

Abstract

In this paper, we introduce a fast Fourier-Galerkin method for solving boundary integral equations on torus-shaped surfaces, which are diffeomorphic to a torus. We analyze the properties of the integral operator's kernel to derive the decay pattern of the entries in the representation matrix. Leveraging this decay pattern, we devise a truncation strategy that efficiently compresses the dense representation matrix of the integral operator into a sparser form containing only $\mathcal{O}(N\ln^2 N)$ nonzero entries, where $N$ denotes the degrees of freedom of the discretization method. We prove that this truncation strategy achieves a quasi-optimal convergence order of $\mathcal{O}(N^{-p/2}\ln N)$, with $p$ representing the degree of regularity of the exact solution to the boundary integral equation. Additionally, we confirm that the truncation strategy preserves stability throughout the solution process. Numerical experiments validate our theoretical findings and demonstrate the effectiveness of the proposed method.

A Fast Fourier-Galerkin Method for Solving Boundary Integral Equations on Torus-Shaped Surfaces

Abstract

In this paper, we introduce a fast Fourier-Galerkin method for solving boundary integral equations on torus-shaped surfaces, which are diffeomorphic to a torus. We analyze the properties of the integral operator's kernel to derive the decay pattern of the entries in the representation matrix. Leveraging this decay pattern, we devise a truncation strategy that efficiently compresses the dense representation matrix of the integral operator into a sparser form containing only nonzero entries, where denotes the degrees of freedom of the discretization method. We prove that this truncation strategy achieves a quasi-optimal convergence order of , with representing the degree of regularity of the exact solution to the boundary integral equation. Additionally, we confirm that the truncation strategy preserves stability throughout the solution process. Numerical experiments validate our theoretical findings and demonstrate the effectiveness of the proposed method.
Paper Structure (8 sections, 11 theorems, 82 equations, 2 figures, 2 tables)

This paper contains 8 sections, 11 theorems, 82 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. A fast Fourier–Galerkin method
  3. Analyzing the entries in $\mathbf{K}_N$
  4. Uniformly analytic extension
  5. Decay pattern of ${K}_{\mathbf{k},\mathbf{l}}$
  6. Stability and convergence analysis
  7. Numerical experiments
  8. Conclusions

Key Result

theorem 1

Let $q>0$. Then there exists a positive constant $c$ such that for all $n \in\mathbb{N}$, $\mathcal{N} (\widetilde{\mathbf{K}}_N)\leq c N\ln^2 N$, where $c$ only depends on $q$, and $N=(2n+1)^2$ is the order of $\widetilde{\mathbf{K}}_N$.

Figures (2)

  • Figure 1: (a) Modulus values of the entries in $\mathbf{K}_{121}$; (b) Modulus values of the entries on the main diagonal and anti-diagonal of $\mathbf{K}_{121}$; (c) Distribution of nonzero entries of $\widetilde{\mathbf{K}}_{121}$.
  • Figure 2: The bagel-shaped surface (left) and the cruller surface (right)

Theorems & Definitions (22)

  • theorem 1
  • proof
  • lemma thmcounterlemma
  • proof
  • theorem 2
  • proof
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • proof
  • ...and 12 more