Discontinuity Analysis and Semi-Analytic Spectral Approximation for the Nonlocal Poisson Equation
We study a nonlocal Poisson problem with discontinuous source term and analyze how the regularity of the integral kernel determines the discontinuity structure of the corresponding solution. Under general assumptions on compactly supported integrable kernels, we show that jump discontinuities in the source term are inherited by the solution. We then identify two principal mechanisms governing higher-order regularity: singular behavior of the kernel at the origin and jump discontinuities of the kernel, or of its derivatives, at the horizon endpoints. Singularities at the origin lead to blow-up of certain derivatives of the solution at the source discontinuity, while jumps at the horizon generate cascades of derivative discontinuities at translated locations. These phenomena occur for kernels commonly used in peridynamic-type models. By contrast, compactly supported \(C^\infty\) kernels do not generate derivative blow-up or cascading losses of regularity, and in this case the source term and the solution have equivalent piecewise smooth regularity. Motivated by this analysis, we develop a semi-analytic spectral method for the accurate numerical treatment of discontinuous nonlocal problems. The method uses successive smoothing transformations and explicitly constructed correction functions to convert the original problem into an auxiliary problem with improved regularity. A spectral solver is then applied to the smoothed problem, and the approximation to the original solution is recovered by adding back the analytic corrections. Numerical experiments show substantial gains in accuracy and convergence, demonstrating that the method effectively mitigates the loss of accuracy caused by discontinuities and Gibbs oscillations while retaining the efficiency of spectral methods.