A posteriori error control for variational inequalities with linear constraints in an abstract framework

This paper proposes a posteriori error control for the discretization of variational inequalities with linear constraints in an abstract framework. The central aspect is the discussion of the error contributions representing the non-penetration, non-conformity and complementarity conditions, which are typically given by some cut-off functions. Replacing the standard cut-off functions with the minimizers of a weighted functional enables the derivation of reliable and, in particular, efficient a posteriori error estimates. The abstract findings are applied to the obstacle problem as well as to the simplified Signorini problem, where higher-order finite elements are used to provide appropriate discretization spaces. Numerical experiments show that the error estimates based on this new approach have (nearly) constant efficiency indeces and reflect the expected order of convergence when uniform h-refinements are applied. Moreover, they can be used to steer adaptive schemes in order to improve the order of convergence. The numerical results are compared with estimates resulting from the standard cut-off functions. © 2021 Journal of Applied and Numerical Optimization