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

TL;DR

This paper introduces a new quadratic finite element method enriched with bubble functions for solving the 3D elliptic obstacle problem, achieving optimal convergence and enabling adaptive mesh refinement.

Contribution

It develops the first quadratic finite element method with bubble enrichment for 3D obstacle problems, including error estimates and adaptive algorithms.

Findings

Optimal convergence demonstrated through error estimates.

Effective adaptive mesh refinement based on a posteriori estimates.

Numerical experiments confirm theoretical results.

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.