Pointwise a posteriori error control for elliptic obstacle problems

Numer. Math. Online DOI 10.1007/s00211-002-0411-3, Dec. 2002.

Ricardo H. Nochetto
Department of Mathematics
University of Maryland, College Park
College Park, MD 20742, USA


Kunibert G. Siebert
Institut fuer Angewandte Mathematik
Hermann-Herder-Str. 10, 79104 Freiburg, Germany


Andreas Veeser
Dipartimento di Matematica
Universita degli Studi di Milano
Via C. Saldini 50, 20133 Milano, Italy



We consider a finite element method for the elliptic obstacle problem over polyhedral domains in $\R^d$, which enforces the unilateral constraint solely at the nodes. We derive novel optimal upper and lower a posteriori error bounds in the maximum norm irrespective of mesh fineness and the regularity of the obstacle, which is just assumed to be H\"older continuous. They exhibit optimal order and localization to the non-contact set. We illustrate these results with simulations in 2d and 3d showing the impact of localization in mesh grading within the contact set along with quasi-optimal meshes.