A multiscale method for nonlocal mechanics and diffusion and for the approximation of discontinuous functions

A multiscale implementation of hybrid continuous/discontinuous finite element discretizations of nonlocal models for mechanics and diffusion in two dimensions is developed. The implementation features adaptive mesh refinement based on the detection of defects and results in an abrupt transition between refined elements that contain defects and unrefined elements free of defects. An additional difficulty overcome in the implementation is the design of accurate quadrature rules for stiffness matrix construction that are valid for any combination of the grid size and horizon parameter, the latter being the extent of nonlocal interactions. As a result, the methodology developed can attain optimal accuracy at very modest additional costs relative to situations for which the solution is smooth. Portions of the methodology can also be used for the optimal approximation, by piecewise linear polynomials, of given functions containing discontinuities. Several numerical examples are provided to illustrate the efficacy of the multiscale methodology.