Two Numerical Schemes Via an Auxiliary Function for Nonlocal Problems With Integrable Kernels

ABSTRACT In this study, we explore volume constrained nonlocal problem with integrable kernels. A central challenge addressed is the discontinuity of true solutions across the boundary, posing difficulties for designing efficient numerical methods. To address this issue, we introduce an auxiliary function which serves as the solution of an auxiliary problem. This approach effectively transforms the original problem to a new nonlocal problem with a true solution which is continuous across the boundary. We propose two numerical schemes to solve the new nonlocal problem: one is exclusively applicable to one‐dimensional case, while the other is suitable for general dimensional cases. Experimental results show that both schemes exhibit optimal convergence order under mild condition.