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A decoupled meshless Nyström scheme for 2D Fredholm integral equations of the second kind with smooth kernels

Bruno Degli Esposti, Alessandra Sestini

TL;DR

This work addresses high-order numerical solutions for Fredholm integral equations of the second kind with smooth kernels on 2D domains. It introduces a decoupled Nyström method that separates solution nodes from quadrature nodes and employs a reconstruction operator to map between these sets, allowing the system size to be decoupled from the quadrature density. The authors prove that the overall convergence rate is the minimum of the quadrature and reconstruction orders and establish convergence via collectively compact operator theory. Numerical experiments with Gaussian and nonpositive kernels validate the theory and demonstrate efficiency, including extensions to nonlinear FIEs.

Abstract

The Nyström method for the numerical solution of Fredholm integral equations of the second kind is generalized by decoupling the set of solution nodes from the set of quadrature nodes. The accuracy and efficiency of the new method is investigated for smooth kernels and complex 2D domains using recently developed moment-free meshless quadrature formulas on scattered nodes. Compared to the classical Nyström method, our variant has a clear performance advantage, especially for narrow kernels. The decoupled Nyström method requires the choice of a reconstruction scheme to approximate values at quadrature nodes from values at solution nodes. We prove that, under natural assumptions, the overall order of convergence is the minimum between that of the quadrature scheme and of the reconstruction scheme.

A decoupled meshless Nyström scheme for 2D Fredholm integral equations of the second kind with smooth kernels

TL;DR

This work addresses high-order numerical solutions for Fredholm integral equations of the second kind with smooth kernels on 2D domains. It introduces a decoupled Nyström method that separates solution nodes from quadrature nodes and employs a reconstruction operator to map between these sets, allowing the system size to be decoupled from the quadrature density. The authors prove that the overall convergence rate is the minimum of the quadrature and reconstruction orders and establish convergence via collectively compact operator theory. Numerical experiments with Gaussian and nonpositive kernels validate the theory and demonstrate efficiency, including extensions to nonlinear FIEs.

Abstract

The Nyström method for the numerical solution of Fredholm integral equations of the second kind is generalized by decoupling the set of solution nodes from the set of quadrature nodes. The accuracy and efficiency of the new method is investigated for smooth kernels and complex 2D domains using recently developed moment-free meshless quadrature formulas on scattered nodes. Compared to the classical Nyström method, our variant has a clear performance advantage, especially for narrow kernels. The decoupled Nyström method requires the choice of a reconstruction scheme to approximate values at quadrature nodes from values at solution nodes. We prove that, under natural assumptions, the overall order of convergence is the minimum between that of the quadrature scheme and of the reconstruction scheme.
Paper Structure (8 sections, 9 theorems, 92 equations)

This paper contains 8 sections, 9 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Fredholm integral equations of the second kind
  3. Equilibrium states of Fredholm integro-differential equations
  4. The Nyström method
  5. Quadrature schemes and the classical Nyström method
  6. A decoupled variant of the Nyström method
  7. Appendix on continuously differentiable functions
  8. Appendix on quadrature schemes

Key Result

Proposition 3.3

With reference to the classical Nyström method eq:nystrom-classical applied to the Fredholm integral equation of the second kind eq:fie, suppose that Then, for all sufficiently small $h > 0$, the linear system eq:nystrom-classical has a unique solution, the condition number of the system matrix is bounded uniformly with respect to $h$, and there exists a constant $C > 0$ independent of $h$ and $u

Theorems & Definitions (26)

  • Definition 3.1
  • Definition 3.2
  • Proposition 3.3
  • proof
  • Definition 3.4
  • Definition 3.5
  • Definition 3.6
  • Proposition 3.7
  • proof
  • Definition 3.8
  • ...and 16 more