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

Abstract: 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 {\em a priori} error estimate is presented.