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Center for Nonlinear Studies |
We develop a vector calculus for nonlocal operators that mimics the classical differential vector calculus. Included are the definition of nonlocal divergence, gradient, and curl operators and derivations of nonlocal Gauss and Stokes theorems and Green's identities. Through appropriate limiting processes, relations between the nonlocal operators and their differential counterparts are established. The nonlocal calculus is 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 including discontinuous Galerkin methods for nonlocal problems, in particular focusing on solutions containing jump discontinuities; in this setting, nonlocal modeling can lead to optimally accurate approximations.
Host: COSIM, Todd Ringler,ringler@lanl.gov