Residual type a posteriori error estimates for elliptic obstacle problems

Abstract

A posteriori error estimators of residual type are derived for piecewise linear finite element approximations to elliptic obstacle problems. An instrumental ingredient is a new interpolation operator which requires minimal regularity, exhibits optimal approximation properties and preserves positivity. Both upper and lower bounds are proved and their optimality is explored with several examples. Sharp a priori bounds for the a posteriori estimators are given, and extensions of the results to double obstacle problems are briefly discussed.Mathematics Subject Classification (1991): 65N15, 65N30; 41A05, 41A29, 41A36

A finite element discretization of the contact between two membranes

“…Estimate (7.13) is also optimal but estimate (7.15) is not because of the power 1/2 on the norm σ − σ h H 1 (ωe) . However the same lack of optimality already appears in [14], Section 4, for a similar problem, see also [6], Remark 4.7. Nevertheless, since these last estimates are fully local, it can be thought that the η K and η e provide a good representation of the local error and hence are an efficient tool for mesh adaptivity.…”

“…After the pioneering paper [1] by Ainsworth et al, a substantial work has been performed on the a posteriori analysis of variational inequalities, see, e.g., [6,16,20] and the references therein. We follow the approach of Hild and Nicaise [14] since they also consider a mixed problem coupling a variational equality and an inequality.…”

Averaging techniques yield reliable a posteriori finite element error control for obstacle problems

“…While a posteriori error control and adaptive mesh design is well established for (elliptic) partial di erential equations AO, BSt, EEHJ, V], their exploitation for variational inequalities started very recently BSu,CN,LLT,V1,V2]. Amongst the a posteriori error estimation techniques are averaging schemes rstly justi ed by super-convergence properties on structured grids with symmetry properties.…”

A posteriorierror analysis for parabolic variational inequalities

“…We point out that failure to recognize the importance of σ(u) leads to a global upper bound of the error but not to a global lower bound [8]; overestimation is thus possible. This issue was first addressed for elliptic variational inequalities by Veeser [23] and further improved by Fierro and Veeser [11] in H 1 (Ω).…”