Nonconforming FEM for the obstacle problem

The main motivation for the application of the Crouzeix-Raviart nonconforming finite element method (NCFEM) to the obstacle problem in this paper is that it allows for fully computable guaranteed lower bounds of the energy and so for simple a posteriori error control. Afurther fully computable and guaranteed upper error bound follows from Braess' work, extended to the Crouzeix-Raviart NCFEM. This error bound competes with the error control from the lower energy bounds. Both a posteriori estimates are efficient with respect to the total error. The paper circumvents variational crimes through a medius analysis and the design of conforming companions. This leads to an improved a priori error analysis for the NCFEMs under minimal regularity assumptions on polyhedral domains. Numerical evidence supports the a priori convergence analysis and confirms guaranteed error control with moderate efficiency indices for uniform and adaptive mesh refinement.