EFFICIENT DISCRETE LAGRANGE MULTIPLIERS IN THREE FIRST-ORDER FINITE ELEMENT DISCRETIZATIONS FOR THE A POSTERIORI ERROR CONTROL IN AN OBSTACLE PROBLEM

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.