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.

翻译：本文通过将求解节点集与积分节点集解耦，推广了用于数值求解第二类Fredholm积分方程的Nyström方法。针对光滑核与复杂二维区域，采用近期发展的基于散乱节点的无矩无网格积分公式，研究了新方法的精度与效率。与经典Nyström方法相比，本改进方案展现出明显的性能优势，尤其适用于窄核情形。解耦Nyström方法需选择重构方案以根据求解节点值逼近积分节点值。我们证明，在自然假设条件下，整体收敛阶是积分方案与重构方案收敛阶中的较小者。