Abstract
Fractional Laplacian equations serve as a pivotal mathematical framework for modeling anomalous diffusion, non-local transport phenomena, and long-range particle interactions. However, the numerical resolution of the integral fractional Laplacian $(-\Delta)^s$ for $s \in (0,1)$ poses formidable computational hurdles due to the non-local nature of the operator, which induces fully populated, dense stiffness matrices upon discretization, exacerbated by geometric complexities on irregular bounded domains. In this paper, we develop a novel wavelet-based multigrid method tailored for non-local fractional diffusion problems posed on arbitrary two-dimensional geometries. By constructing a multi-resolution biorthogonal wavelet basis adapted to domain embeddings via diffuse domain approximations, we compress the non-local operator matrix to quasi-sparse representations with controlled error bounds. A tailored geometric-algebraic multigrid cycle is designed, utilizing wavelet scale-dependent smoothers and energy-orthogonal inter-grid transfer operators. Rigorous theoretical analysis demonstrates that the condition number of the wavelet-preconditioned operator remains bounded independently of the discretization scale, leading to mesh-independent multigrid convergence. Numerical experiments on complex domains—including L-shaped, multiply-connected, and fractal-boundary geometries—confirm that the proposed methodology reduces the computational complexity from $\mathcal{O}(N^3)$ of direct dense solvers to $\mathcal{O}(N \log^2 N)$, while achieving robust spectral error convergence across the entire spectrum of the fractional exponent $s$.
