Department of Scientific Computing
Florida State University
mgunzburger@fsu.edu
Abstract: We study nonlocal models for diffusion and
mechanics, including the nonlocal, spatial derivative free peridynamics model
for solid mechanics. Our focus is on the analysis of well posedness and on
finite element methods. Both rely on a vector calculus we have developed for
nonlocal operators that mimics the classical differential vector
calculus. Included are the definitions of nonlocal divergence, gradient, and
curl operators and the derivation of nonlocal integral theorems and
identities. The nonlocal calculus is then applied to nonlocal diffusion and
mechanics problems; in particular, strong and weak formulations of these
problems are considered and analyzed, showing, for example, that unlike
elliptic partial differential equations, these problems do not necessary
result in the smoothing of data. Finally, we briefly consider finite element
methods for nonlocal problems, focusing on solutions containing jump
discontinuities; in this setting, nonlocal models can lead to optimally
accurate approximations.