A numerical method is proposed for a class of stochastic control problems including singular behavior. This method solves an infinite-dimensional linear program equivalent to the stochastic control problem using a finite element type approximation, which results in a solvable finite-dimensional program. The discretization scheme as well as the necessary assumptions are discussed, and a detailed convergence analysis for the discretization scheme is given. Its performance is illustrated by two examples featuring a long-term average cost criterion.