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Carnegie Mellon Center for Nonlinear Analysis
A nonlocal Vector Calculus and the Analysis and Approximation of Nonlocal Models for Diffusion and Mechanics


Max Gunzburger
Department of Scientific Computing
Florida State University

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.

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