  In this paper, we study the efficient numerical integration of functions with
sharp gradients and cusps. An adaptive integration algorithm is presented that
systematically improves the accuracy of the integration of a set of functions.
The algorithm is based on a divide and conquer strategy and is independent of
the location of the sharp gradient or cusp. The error analysis reveals that for
a $C^0$ function (derivative-discontinuity at a point), a rate of convergence
of $n+1$ is obtained in $R^n$. Two applications of the adaptive integration
scheme are studied. First, we use the adaptive quadratures for the integration
of the regularized Heaviside function---a strongly localized function that is
used for modeling sharp gradients. Then, the adaptive quadratures are employed
in the enriched finite element solution of the all-electron Coulomb problem in
crystalline diamond. The source term and enrichment functions of this problem
have sharp gradients and cusps at the nuclei. We show that the optimal rate of
convergence is obtained with only a marginal increase in the number of
integration points with respect to the pure finite element solution with the
same number of elements. The adaptive integration scheme is simple, robust, and
directly applicable to any generalized finite element method employing
enrichments with sharp local variations or cusps in $n$-dimensional
parallelepiped elements.
