Efficient Discrete Lagrange Multipliers in three first-order finite element discretizations for the A Posteriori Error Control in an Obstacle Problem

Abstract

The efficient design of a discrete Lagrange multiplier is a key ingredient in the a posteriori error analysis for the obstacle problem. Affirmative examples exist for three different first-order finite element methods (FEMs), namely, the $P_1$ conforming Courant, the $P_1$ nonconforming Crouzeix--Raviart, and the lowest-order mixed Raviart--Thomas. With those discrete Lagrange multipliers, a general reliable and efficient a posteriori error analysis for the error in the energy norm of the displacement variables applies to all those discretizations for affine obstacles under minimal assumptions.