Analysis and Approximation of Nonlocal Diffusion Problems with Volume Constraints

We exploit a recently developed nonlocal vector calculus to provide a variational analysis for a general class of nonlocal diffusion problems given by a linear integral equation on bounded domains in R. The ubiquity of the nonlocal operator is illustrated by a number of applications ranging from continuum mechanics to graph theory. These applications elucidate different interpretations of the operator and the governing equation. A probabilistic perspective explains that the nonlocal operator corresponds to the infinitesimal generator for a symmetric jump process. Sufficient conditions on the kernel of the operator and the notion of volume constraints lead to a well-posed problem. The volume constraints are a proxy for boundary conditions that may not be defined for the given kernel. In particular, we demonstrate for a general class of kernels that the nonlocal operator is a mapping between a constrained subspace of a fractional Sobolev subspace and its dual. We also demonstrate for some other kernels the operator’s inverse does not smooth but does correspond to diffusion. The impact of our analyses is that both a continuum analysis and a numerical method for the modeling of anomalous diffusion on bounded domains in R are provided. The analytical framework also allows us to consider finite dimensional approximations using both discontinuous or continuous Galerkin methods that are conforming for the nonlocal diffusion equation; error and condition number estimates are derived. The nonlocal vector calculus enables striking analogies to be drawn with the problem of classical diffusion including a notion of nonlocal flux.