Optimally convergent (with respect to the regularity) quadratic finite element method for two dimensional obstacle problem on simplicial meshes is studied in (Brezzi, Hager, Raviart, Numer. Math, 28:431-443, 1977). There was no analogue of a quadratic finite element method on tetrahedron meshes for three dimensional obstacle problem. In this article, a quadratic finite element enriched with element-wise bubble functions is proposed for the three dimensional elliptic obstacle problem. A priori error estimates are derived to show the optimal convergence of the method with respect to the regularity. Further a posteriori error estimates are derived to design an adaptive mesh refinement algorithm. Numerical experiment illustrating the theoretical result on a priori error estimate is presented.1991 Mathematics Subject Classification. 65N30, 65N15.

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.

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We propose a new and simpler residual based a posteriori error estimator for finite element approximation of the elliptic obstacle problem. The results in the article are two fold. Firstly, we address the influence of the inhomogeneous Dirichlet boundary condition in a posteriori error control of the elliptic obstacle problem. Secondly by rewriting the obstacle problem in an equivalent form, we derive simpler a posteriori error bounds which are free from min/max functions. To accomplish this, we construct a post-processed solution ũh of the discrete solution u h which satisfies the exact boundary conditions although the discrete solution u h may not satisfy. We propose two post processing methods and analyze them. We remark that the results known in the literature are either for the homogeneous Dirichlet boundary condition or that the estimator is only weakly reliable in the case of inhomogeneous Dirichlet boundary condition.