  Numerical solution of nonlocal constrained value problems with integrable
kernels are considered. These nonlocal problems arise in nonlocal mechanics and
nonlocal diffusion. The structure of the true solution to the problem is
analyzed first. The analysis leads naturally to a new kind of discontinuous
Galerkin method that efficiently solve the problem numerically. This method is
shown to be asymptotically compatible. Moreover, it has optimal convergence
rate for one dimensional case under very weak assumptions, and almost optimal
convergence rate for two dimensional case under mild assumptions.
