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We present the first systematic work for deriving a posteriori errorestimates for general non-polynomial basis functions in an interior penaltydiscontinuous Galerkin (DG) formulation for solving second order linear PDEs.Our residual type upper and lower bound error estimates measure the error inthe energy norm. The main merit of our method is that the method isparameter-free, in the sense that all but one solution-dependent constantsappearing in the upper and lower bound estimates are explicitly computable bysolving local eigenvalue problems, and the only non-computable constant can bereasonably approximated by a computable one without affecting the overalleffectiveness of the estimates in practice. As a side product of ourformulation, the penalty parameter in the interior penalty formulation can beautomatically determined as well. We develop an efficient numerical procedureto compute the error estimators. Numerical results for a variety of problems in1D and 2D demonstrate that both the upper bound and lower bound are effective.

A posteriori error estimates for discontinuous Galerkin methods using non-polynomial basis functions. Part I: Second order linear PDE