A posteriori error estimates for discontinuous Galerkin methods using non-polynomial basis functions. I: Second order linear PDE. (Q2820349)

The authors present a systematic work to derive a posteriori error estimates for general non-polynomial basis functions in an interior penalty discontinuous Galerkin (DG) formulation for solving second-order linear partial differential equations. The residual type upper and lower bound error estimates measure the error in the energy norm. The method presented is parameter-free, in the sense that all but one solution-dependent constants appearing in the upper and lower bound estimates are explicitly computable by solving local eigenvalue problems, and the only non-computable constant can be reasonably approximated by a computable one without affecting the overall effectiveness of the estimates in practice. The penalty parameter in the interior penalty formulation can be automatically determined. An efficient numerical procedure to compute the error estimators is developed. Numerical results for several problems in 1D and 2D demonstrate that both the upper bound and lower bound are effective.