A least-squares Fourier frame method for nonlocal diffusion models on arbitrary domains

We introduce a least-squares Fourier frame method for solving nonlocal diffusion models with Dirichlet volume constraint on arbitrary domains. The mathematical structure of a frame rather than a basis allows using a discrete least-squares approximation on irregular domains and imposing non-periodic boundary conditions. The method has inherited the one-dimensional integral expression of Fourier symbols of the nonlocal diffusion operator from Fourier spectral methods for any d spatial dimensions. High precision of its solution can be achieved via a direct solver such as pivoted QR decomposition even though the corresponding system is extremely ill-conditioned, due to the redundancy in the frame. The extension of AZ algorithm improves the complexity of solving the rectangular linear system to $O(N{\log }^{2}N)$ for 1d problems and $O(N^2{\log }^{2}N)$ for 2d problems, compared with $O\left(N^3\right)$ of the direct solvers, where N is the number of degrees of freedom. We present ample numerical experiments to show the flexibility, fast convergence and asymptotical compatibility of the proposed method.