A posteriori error control of discontinuous Galerkin methods for elliptic obstacle problems (Q2871177)

The paper deals with a posteriori error estimates for various discontinuous Galerkin (DG) methods for elliptic obstacle problems. A priori error estimates for elliptic variational inequalities of the first and the second kind are considered by \textit{F. Wang} et al. [SIAM J. Numer. Anal. 48, No. 2, 708--733 (2010; Zbl 1214.65039)]. These authors used piecewise linear and quadratic discontinuous finite element spaces. In the present paper, the error estimator is derived by the help of a nonlinear smoothing function. The error estimator is comparable with the known estimator for conforming finite element methods. The error estimator for DG methods involves a discrete Lagrange multiplier. Under the assumption of the Poincaré inequality it is proved that this Lagrange multiplier is stable uniformly on non-uniform meshes. Applications of the results to various DG methods are discussed. Numerical results demonstrating the performance of the error estimator are given. For instance, an example of \textit{S. Bartels} and \textit{C. Carstensen} [Numer. Math. 99, No. 2, 225--249 (2004; Zbl 1063.65050)] is studied. Conclusions and future problems in the topic are mentioned at the end.