The main emphasis of this thesis is on developing and implementing linear and quadratic finite element methods for 3-dimensional (3D) elliptic obstacle problems. The study consists of a priori and a posteriori error analysis of conforming as well as discontinuous Galerkin methods on a 3D domain. The work in the thesis also focuses on constructing reliable and efficient error estimator for elliptic obstacle problem with inhomogenous boundary data on a 2D domain.

Finally, a MATLAB implementation of uniform mesh refinement for a 3D domain is also discussed. In this talk, we first present a quadratic finite element method for three dimensional ellipticobstacle problem which is optimally convergent (with respect to the regularity). We derive a priorierror estimates to show the optimal convergence of the method with respect to the regularity, forthis we have enriched the finite element space with element-wise bubble functions. Further, aposteriori error estimates are derived to design an adaptive mesh refinement algorithm. The result on a priori estimate will be illustrated by a numerical experiment. Next, we discuss on two newly proposed discontinuous Galerkin (DG) finite element methods for the elliptic obstacle problem.Using the localized behavior of DG methods, we derive a priori and a posteriori error estimates forlinear and quadratic DG methods in dimension 2 and 3 without the addition of bubble functions.We consider two discrete sets, one with integral constraints (motivated as in the previous work)and another with point constraints at quadrature points. The analysis is carried out in a unified setting which holds for several DG methods with variable polynomial degree. We then proposea new and simpler residual based a posteriori error estimator for finite element approximationof the elliptic obstacle problem. The results here are two fold. Firstly, we address the influenceof 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 ˜uh of the discrete solution uh which satisfies the exact boundaryconditions although the discrete solution uh may not satisfy. We propose two post processing methods and analyse 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. Finally, we discuss a uniform mesh refinement algorithm for a 3D domain. Starting with orientation of a face of the tetrahedron and orientation of the tetrahedron, we discuss the ideas for nodes to element connectivity and red-refinement of a tetrahedron. We present conclusions and possible extensions for the future works.