We present the first systematic work for deriving a posteriori error estimates for general non-polynomial basis functions in an interior penalty discontinuous Galerkin(DG)formulation.The main merit of our residual type upper and lower bound error estimates is that the method is almost parameterfree,in the sense that all but one solution-dependent constants are explicitly computable.We develop an efficient numerical procedure to compute the error estimators.(Joint work with Benjamin Stamm)