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

TL;DR

This paper develops a systematic approach for deriving a posteriori error estimates in energy norm for second order linear PDEs solved with discontinuous Galerkin methods using non-polynomial basis functions, with explicit constants and automatic penalty parameter selection.

Contribution

It introduces a parameter-free a posteriori error estimation method for DG methods with non-polynomial basis functions, including explicit computable constants and automatic penalty parameter determination.

Findings

Error bounds are effective in 1D and 2D numerical tests.

Constants in error estimates are explicitly computable or approximable.

The method is applicable to general non-polynomial basis functions.

Abstract

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 for solving second order linear PDEs. Our residual type upper and lower bound error estimates measure the error in the energy norm. The main merit of our method is that the method 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. As a side product of our formulation, the penalty parameter in the interior penalty formulation can be automatically determined as well. We develop an efficient numerical procedure to compute the error estimators. Numerical results for a variety of problems in 1D and 2D demonstrate that both the upper bound and lower bound are effective.