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

TL;DR

The paper analyzes a class of nonlocal elliptic equations of the form $- abla^2 u + \lambda G(u) = f$ on bounded domains with homogeneous Dirichlet boundary through a variational lens, proving existence, uniqueness, and positivity of weak solutions in $H_0^1(\Omega)$ via the Lax–Milgram theorem. It defines the nonlocal operator $G$ by a Gaussian kernel and establishes its boundedness and self-adjointness under standard kernel assumptions, then introduces extension/padding operators to ensure boundary-consistent discretization. A finite-difference discretization combined with a fixed-point iteration is developed, with explicit stability and convergence conditions tying the step size $\tau$, regularization parameter $\lambda$, and the Poincaré constant $C_P$. Numerical experiments on bounded domains validate contraction and robustness across padding schemes, demonstrating monotone residual decay and accurate nonlocal behavior. The work delivers a rigorous, padding-aware framework for solving nonlocal elliptic problems with Gaussian kernels and lays groundwork for extensions to non-symmetric kernels and higher-dimensional settings.

Abstract

This paper investigates the mathematical properties and numerical approximation of a class of nonlocal elliptic partial differential equations of the form \begin{equation*} -Δu + λ\, G(u) = f, \end{equation*} where $Δ$ denotes the Laplacian, $λ> 0$ 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 $H_0^1(Ω)$ 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 $f \in L^2(Ω)$. 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.

Figures (2)

• Figure 1: Three-dimensional visualization of the numerical solution $u(x,y)$ (solid surface) and the corresponding supersolution $w(x,y)$ (wireframe) over the domain $\Omega = [0,1] \times [0,1]$.
• Figure 2: Convergence of the fixed-point iteration: residual norm $\|r^{(k)}\|$ versus iteration number for various parameter configurations. The monotonic decay validates the contraction conditions established in Theorem \ref{['cont']}.

Theorems & Definitions (13)

• Proposition 1
• Proposition 2: Lax-Milgram
• Proposition 3: Boundedness and Self-adjointness
• proof
• Proposition 4: Nonlocal Positivity Principle
• proof
• Theorem 1
• proof
• Proposition 5: First variation and Euler--Lagrange equation
• proof