Finite Element Method for Solving the Screened Poisson Equation with a Delta Function

This paper presents a Finite Element Method (FEM) framework for solving the screened Poisson equation with a Dirac delta function as the forcing term. The singularity introduced by the delta function poses challenges for standard numerical methods, particularly in higher dimensions. To address this, we employ integrated Legendre basis functions, which yield sparse and structured system matrices characterized by a Banded-Block-Banded-Arrowhead (B3-Arrowhead) form. In one dimension, the resulting linear system can be solved directly. In two and three dimensions, the equation can be efficiently solved using a generalized Alternating Direction Implicit (ADI) method combined with reverse Cholesky factorization. Numerical results in 1D, 2D, and 3D confirm that the method accurately captures the localized impulse response and reproduces the expected Green’s function behavior. The proposed approach offers a robust and scalable solution framework for partial differential equations with singular source terms and has potential applications in physics, engineering, and computational science.