Non-conforming Finite Element Methods for the Obstacle Problem

Abstract. In an obstacle problems with an affine obstacle, homogeneous Dirichlet boundary conditions, and standard regularity assumptions, the Crouzeix-Raviart non-conforming finite element method (FEM) allows for linear convergence as the maximal mesh-size approaches zero. The residual-based a posteriori error analysis leads to reliable and efficient control over the error with explicit constants. It involves the design of a new discrete Lagrange multiplier and allows for the computation of a guaranteed upper error bound. A novel energy control for nonconforming FEMs lead to a computable guaranteed lower bound for the minimal energy. The paper presents numerical experiments to investigate the theoretical results empirically and so to explore the possibilities of the non-conforming finite element method with respect to adaptive mesh refinement in practice.