A posteriori error estimator and error control for contact problems

Abstract

Abstract. In this paper, we consider two error estimators for one-body contact problems. The first error estimator is defined in terms of H(div)-conforming stress approximations and equilibrated fluxes while the second is a standard edge-based residual error estimator without any modification with respect to the contact. We show reliability and efficiency for both estimators. Moreover, the error is bounded by the first estimator with a constant one plus a higher order data oscillation term plus a term arising from the contact that is shown numerically to be of higher order. The second estimator is used in a control-based AFEM refinement strategy, and the decay of the error in the energy is shown. Several numerical tests demonstrate the performance of both estimators.

A posteriori estimators for obstacle problems by the hypercircle method

“…It is therefore clear that it does not spoil the efficiency of the error estimate. We note that Weiss and Wohlmuth [15] observed a similar phenomenon when they considered inequality constraints on the boundary of the domain Ω. On the other hand, Repin [12] considered the hypercircle method without the regularity assumption of the active set, and his result is closer to the classical estimates than to our result with the hypercircle method.…”

On the unilateral contact between membranes. Part 2: a posteriori analysis and numerical experiments

“…After the pioneering works [13] by Hlaváček, Haslinger, Nečas, and Lovíšek (see Theorem 4.2 in this book) and [1] by Ainsworth, Oden, and Lee, a huge amount of work has been performed on the a posteriori analysis of variational inequalities, see, e.g., [2], [5], [11], [14], [16], [19], [20], and the references therein. It can be noted that, in [11], a mixed problem coupling a variational equality and an inequality is also considered.…”

A posteriori error estimates of discontinuous Galerkin methods for the Signorini problem

“…The work related to the obstacle problem can be found in [16,17,18,19,28,33,46] and in [5,10,23,31,44]. Related to the work on the Signorini problem, we refer to [4,28,38,42,30] for a priori error analysis and to [9,34,35,48] for a posteriori analysis. The analysis in [35,48] is for a mixed formulation of the Signorini problem introducing a Lagrange multiplier and the analysis in [9,34] is for a simplified Signorini problem.…”

“…Related to the work on the Signorini problem, we refer to [4,28,38,42,30] for a priori error analysis and to [9,34,35,48] for a posteriori analysis. The analysis in [35,48] is for a mixed formulation of the Signorini problem introducing a Lagrange multiplier and the analysis in [9,34] is for a simplified Signorini problem. Also refer to the recent work in [40] for the a posteriori error control of conforming methods.…”