Well-Posedness and Numerical Approximation of a Class of Nonlocal Elliptic Equations with Gaussian Kernels

Abstract

This paper investigates the mathematical properties and numerical approximation of a class of nonlocal elliptic partial differential equations of the form \begin{equation*} -\Delta u + \lambda \, G(u) = f, \end{equation*} where $\Delta$ denotes the Laplacian, $\lambda > 0$ is a regularization parameter, and $G$ is a nonlocal operator defined by integral convolution with a kernel $K$. We establish the well-posedness of the problem in the Sobolev space $H_0^1(\Omega)$ using the Lax--Milgram theorem, providing rigorous proofs for the existence, uniqueness, and positivity of the weak solution under standard assumptions on the kernel $K$ and the source term $f \in L^2(\Omega)$. For the numerical treatment, we employ a finite difference discretization for the Laplacian and a Gaussian-based approximation for the nonlocal term. We analyze a fixed-point iterative scheme for solving the discrete system and derive explicit conditions for its convergence and stability. Numerical experiments validate the theoretical results, demonstrating the monotonic decay of the residual and the robustness of the approximation scheme on bounded domains with various padding strategies.