Nonlocal Integral Equation Continuum Models: Nonstandard Symmetric Interaction Neighborhoods and Finite Element Discretizations

The book presents the state of the art of nonlocal modeling and discretization and novel analyses of a class of nonstandard nonlocal models. These models have recently become a viable alternative to classical partial differential equations when the latter are unable to capture effects such as discontinuities and multiscale behavior in a system of interest. Because of their integral nature, nonlocal operators allow for the relaxation of regularity requirements on the solution and for capturing multiscale effects and thus have been successfully used in scientific and engineering applications such as diffusion processes, fracture mechanics, heterogeneous material response, subsurface transport, turbulence, and image processing, to name a few. Although the use of nonstandard models is novel, the book provides extensive background and a thorough analysis and description of their discretization methods, enabling it to serve as a gentle and practical introduction to nonlocal modeling for readers who are not familiar with nonlocality.