  In this paper we design efficient quadrature rules for finite element
discretizations of nonlocal diffusion problems with compactly supported kernel
functions. Two of the main challenges in nonlocal modeling and simulations are
the prohibitive computational cost and the nontrivial implementation of
discretization schemes, especially in three-dimensional settings. In this work
we circumvent both challenges by introducing a parametrized mollifying function
that improves the regularity of the integrand, utilizing an adaptive
integration technique, and exploiting parallelization. We first show that the
"mollified" solution converges to the exact one as the mollifying parameter
vanishes, then we illustrate the consistency and accuracy of the proposed
method on several two- and three-dimensional test cases. Furthermore, we
demonstrate the good scaling properties of the parallel implementation of the
adaptive algorithm and we compare the proposed method with recently developed
techniques for efficient finite element assembly.
