Analysis and numerical methods for nonlocal models

Abstract

This dissertation addresses the regularity of solutions for nonlocal diffusion equations over the space of periodic distributions. The spatial operator for the nonlocal diffusion equation is given by a nonlocal Laplace operator with a compactly supported integral kernel. We follow a unified approach based on the Fourier multipliers of the nonlocal Laplace operator, which allows the study of regular and distributional solutions of the nonlocal diffusion equation, as well as integrable and singular kernels, in any spatial dimension. In addition, the results extend beyond operators with singular kernels to nonlocal super-diffusion operators. We present results on the spatial and temporal regularity of solutions in terms of the regularity of the initial data or the diffusion source term. Moreover, solutions of the nonlocal diffusion equation are shown to converge to the solution of the classical diffusion equation for two types of limits: as the spatial nonlocality vanishes or as the singularity of the integral kernel approaches a certain critical singularity that depends on the spatial dimension. Furthermore, we show that, for the case of integrable kernels, discontinuities in the initial data propagate and persist in the solution of the nonlocal diffusion equation. The magnitude of a jump discontinuity is shown to decay over time.

In addition, we present a spectral numerical method for nonlocal equations on bounded domains. These spectral solvers exploit the fact that integration in the nonlocal formulation transforms into multiplication in Fourier space and that nonlocality is decoupled from the grid size. Our approach extends the spectral solvers developed by Alali and Albin (2020) for periodic domains by incorporating the two-dimensional Fourier Continuation (2D-FC) algorithm introduced in Bruno and Paul (2022). We evaluate the performance of the proposed methods on two-dimensional nonlocal Poisson and nonlocal diffusion equations defined on bounded domains. While the regularity of solutions to these equations in bounded settings remains an open problem, we conduct numerical experiments to explore this issue, particularly focusing on studying discontinuities.