A decoupled meshless Nystr\"om scheme for 2D Fredholm integral equations of the second kind with smooth kernels

TL;DR

This paper introduces a decoupled meshless Nyström method for solving 2D Fredholm integral equations of the second kind with smooth kernels, improving accuracy and efficiency over classical methods especially for narrow kernels.

Contribution

It proposes a novel decoupled approach that separates solution and quadrature nodes, enhancing performance and providing convergence analysis for smooth kernels on complex domains.

Findings

The decoupled method outperforms classical Nyström in accuracy and efficiency.

Convergence order is determined by the minimum of quadrature and reconstruction schemes.

Effective for complex 2D domains with smooth kernels.

Abstract

The Nystr\"om 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 Nystr\"om method, our variant has a clear performance advantage, especially for narrow kernels. The decoupled Nystr\"om 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.