  The implementation of the finite element method for linear elliptic equations
requires to assemble the stiffness matrix and the load vector. In general, the
entries of this matrix-vector system are not known explicitly but need to be
approximated by quadrature rules. If the coefficient functions of the
differential operator or the forcing term are irregular, then standard
quadrature formulas, such as the barycentric quadrature rule, may not be
reliable. In this paper we investigate the application of two randomized
quadrature formulas to the finite element method for such elliptic boundary
value problems with irregular coefficient functions. We give a detailed error
analysis of these methods, discuss their implementation, and demonstrate their
capabilities in several numerical experiments.
