Mitigating Edge Singularities in Point-Based Discretization of Integral Equations: A Study on Mixed-Order Discretization ($p$-Refinement) Coupled with Local $h$-Refinement

In this work, we present a framework for efficiently solving sharp-edge geometries in integral equations without explicitly capturing singularities at the edges. The framework is designed for point-based discretization methods, particularly the mixed-order Locally Corrected Nyström (LCN) method and Chebyshev-based Nyström (CBIE) discretization. A hybrid approach is introduced: low-order solutions with local h-refinement are employed at edge elements to maintain accuracy, while high-order LCN or CBIE methods are applied to non-edge elements. This strategy ensures both accuracy and computational efficiency, mitigating the numerical difficulties posed by sharp edges. The results validate that this framework enhances solution accuracy while maintaining numerical stability in sharp-edge geometries.