A Posteriori and A Priori Error Estimates of Quadratic Finite Element Method for Elliptic Obstacle Problem

Abstract: A residual based {\em a posteriori} error estimator is derived for a quadratic finite element method (fem) for the elliptic obstacle problem. The error estimator involves various residuals consisting the data of the problem, discrete solution and a Lagrange multiplier related to the obstacle constraint. {\em A priori} error estimates for the Lagrange multiplier have been derived and further under an assumption that the contact set does not degenerate to a curve in any part of the domain, optimal order {\em a priori} error estimates have been derived whenever the data and the solution are sufficiently regular, precisely, under the sufficient conditions required for quadratic fem in the case of linear elliptic problem. The numerical experiments of adaptive fem for a model problem satisfying the above condition on contact set show optimal order convergence. This demonstrates that the quadratic fem for obstacle problem can exhibit optimal performance.