Two new approaches for solving elliptic obstacle problems using discontinuous Galerkin methods

Abstract

The main aim of this article is to present two new ways to solve the elliptic obstacle problem by using discontinuous Galerkin finite element methods. In Gaddam and Gudi (Comput Methods Appl Math 18:223–236, 2018. https://doi.org/10.1515/cmam-2017-0018), a bubble enriched conforming quadratic finite element method is introduced and analyzed for the obstacle problem in dimension 3. In this article, without adding bubble functions, we derive optimal order (with respect to regularity) a priori error estimates in dimension 2 and 3 using the localized behavior of DG methods. We consider two different discrete sets, one with integral constraints motivated from Gaddam and Gudi (2018) and the other with nodal constraints at quadrature points. We also discuss the reliability and efficiency of a proposed a posteriori error estimator. The analysis is carried out in a unified setting which holds for several DG methods. Numerical results are presented to illustrate the theoretical findings.