A Reliable Residual Based A Posteriori Error Estimator for a Quadratic Finite Element Method for the Elliptic Obstacle Problem

Abstract

A residual based 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. 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 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. 1991 Mathematics Subject Classification. 65N30, 65N15. Key words and phrases. finite element, quadratic fem, a posteriori error estimate, obstacle problem, optimal error estimates, variational inequalities, Lagrange multiplier.

An A Posteriori Analysis of $C^0$ Interior Penalty Methods for the Obstacle Problem of Clamped Kirchhoff Plates

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Discontinuous Galerkin methods for solving a hyperbolic inequality

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