Asymptotically compatible schemes for the approximation of fractional Laplacian and related nonlocal diffusion problems on bounded domains

Abstract

Approximations of solutions of fractional Laplacian equations on bounded domains are considered. Such equations allow global interactions between points separated by arbitrarily large distances. Two approximations are introduced. First, interactions are localized so that only points less than some specified distance, referred to as the interaction radius, are allowed to interact. The resulting truncated problem is a special case of a more general nonlocal diffusion problem. The second approximation is the spatial discretization of the related nonlocal diffusion problem. A recently developed abstract framework for asymptotically compatible schemes is applied to prove convergence results for solutions of the truncated and discretized problem to the solutions of the fractional Laplacian problems. Intermediate results also provide new convergence results for the nonlocal diffusion problem. Special attention is paid to limiting behaviors as the interaction radius increases and the spatial grid size decreases, regardless of how these parameters may or may not be dependent. In particular, we show that conforming Galerkin finite element approximations of the nonlocal diffusion equation are always asymptotically compatible schemes for the corresponding fractional Laplacian model as the interaction radius increases and the grid size decreases. The results are developed with minimal regularity assumptions on the solution and are applicable to general domains and general geometric meshes with no restriction on the space dimension and with data that are only required to be square integrable. Furthermore, our results also solve an open conjecture given in the literature about the convergence of numerical solutions on a fixed mesh as the interaction radius increases.