Supremum-norm a posteriori error control of quadratic discontinuous Galerkin methods for the obstacle problem

TL;DR

This paper develops a supremum-norm a posteriori error estimator for quadratic discontinuous Galerkin methods applied to the elliptic obstacle problem, providing reliable and efficient error control with numerical validation.

Contribution

It introduces a novel a posteriori error estimator for quadratic DG methods in the supremum norm, incorporating integral and nodal constraints, and employs Green's function bounds for analysis.

Findings

Estimator is reliable and efficient in numerical tests

Error bounds are sharp and theoretically justified

Numerical experiments confirm theoretical predictions

Abstract

We perform a posteriori error analysis in the supremum norm for the quadratic discontinuous Galerkin method for the elliptic obstacle problem. We define two discrete sets (motivated by Gaddam, Gudi and Kamana [1]), one set having integral constraints and other one with the nodal constraints at the quadrature points, and discuss the pointwise reliability and efficiency of the proposed a posteriori error estimator. In the analysis, we employ a linear averaging function to transfer DG finite element space to standard conforming finite element space and exploit the sharp bounds on the Green's function of the Poisson's problem. Moreover, the upper and the lower barrier functions corresponding to continuous solution u are constructed by modifying the conforming part of the discrete solution uh appropriately. Finally, numerical experiments are presented to complement the theoretical results.