Memory-efficient implementation of the Heat semi-group (HSG) integral solution for the fractional screened Poisson equation on bounded domains

This paper presents a novel approach to solving the fractional screened Poisson equation on bounded domains using the heat semi-group method. This method involves finding a solution to the fractional problem by integrating the transient diffusion equation solution against a suitable weight function The methodology of the present paper consist in: to solve the pure diffusion equation using Finite Element Method (FEM); then integrate the diffusion solution using Gauss-Legendre quadrature adapted to a uniform time-grid trough first order Lagrangian interpolation. Theoretical error bounds are showed. Additionally, we propose a memory-efficient implementation of the numerical scheme to reduce the memory requirements of the code. Several numerical experiments in one and two dimensions have been run to test the integral solution’s performance and effectiveness.