Bubbles Enriched Quadratic Finite Element Method for the 3D-Elliptic Obstacle Problem

An optimally convergent (with respect to the regularity) quadratic finite element method for the two-dimensional obstacle problem on simplicial meshes is studied in 14]. There was no analogue of a quadratic finite element method on tetrahedron meshes for the 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. A numerical experiment illustrating the theoretical result on a priori error estimates is presented.