  We investigate the numerical approximation of an elliptic optimal control
problem which involves a nonconvex local regularization of the $L^q$-quasinorm
penalization (with $q\in(0,1)$) in the cost function. Our approach is based on
the \emph{difference-of-convex} function formulation, which leads to
first-order necessary optimality conditions, which can be regarded as the
optimality system of an auxiliar convex $L^1$-penalized optimal control
problem.
  We consider piecewise-constant finite element approximation for the controls,
whereas the state equation is approximated using piecewise-linear basis
functions. Then, convergence results are obtained for the proposed
approximation. Under certain conditions on the support's boundary of the
optimal control, we deduce an order of $h^\frac12$ approximation rate of
convergence where $h$ is the associated discretization parameter. We illustrate
our theoretical findings with numerical experiments that show the convergence
behavior of the numerical approximation
