L^2 and L^\infty errors vs. the number of intervals N=1/h for finite element discretizations of a peridynamics model corresponding to a discontinuous exact solution. Left: discontinuous piecewise-constant approximations. Middle: discontinuous piecewise-linear approximations. Right: hybrid approximation using discontinuous piecewise-linears near the discontinuity and continuous piecewise-linears away from the discontinuity. Each figure contains plots for three values of the peridynamic horizon. Note that discontinuous approximations are conforming for the peridynamics model because that model does not contain any spatial derivatives. For the case presented here, they are also optimally accurate.