Error Reduction in Adaptive Finite Element Approximations of Elliptic Obstacle Problems (Journal of Computational Mathematics, 2018)
We consider an adaptive finite element method (AFEM) for obstacle problems associatedwith linear second order elliptic boundary value problems and prove a reductionin the energy norm of the discretization error which leads to R-linear convergence. Thisresult is shown to hold up to a consistency error due to the extension of the discretemultipliers (point functionals) to $H^{−1}$ and a possible mismatch between the continuousand discrete coincidence and noncoincidence sets. The AFEM is based on a residual-typeerror estimator consisting of element and edge residuals. The a posteriori error analysisreveals that the significant difference to the unconstrained case lies in the fact that theseresiduals only have to be taken into account within the discrete noncoincidence set. Theproof of the error reduction property uses the reliability and the discrete local efficiency ofthe estimator as well as a perturbed Galerkin orthogonality. Numerical results are givenillustrating the performance of the AFEM.