Adaptive inexact smoothing Newton method for a nonconforming discretization of a variational inequality

We develop in this work an adaptive inexact smoothing Newton method for a nonconforming discretization of a variational inequality. As a model problem, we consider the contact problem between two membranes. Discretized with the finite volume method, this leads to a nonlinear algebraic system with complementarity constraints. The non-differentiability of the arising nonlinear discrete problem a priori requests the use of an iterative linearization algorithm in the semismooth class like, e.g., the Newton-min. In this work, we rather approximate the inequality constraints by a smooth nonlinear equality, involving a positive smoothing parameter that should be drawn down to zero. This makes it possible to directly apply any standard linearization like the Newton method. The solution of the ensuing linear system is then approximated by any iterative linear algebraic solver. In our approach, we carry out an a posteriori error analysis where we introduce potential reconstructions in discrete subspaces included in H-1(Omega), as well as H(div, Omega)- conforming discrete equilibrated flux reconstructions. With these elements, we design an a posteriori estimate that provides guaranteed upper bound on the energy error between the unavailable exact solution of the continuous level and a postprocessed, discrete, and available approximation, and this at any resolution step. It also offers a separation of the different error components, namely, discretization, smoothing, linearization, and algebraic. Moreover, we propose stopping criteria and design an adaptive algorithm where all the iterative procedures (smoothing, linearization, algebraic) are adaptively stopped; this is in particular our way to fix the smoothing parameter. Finally, we numerically assess the estimate and confirm the performance of the proposed adaptive algorithm, in particular in comparison with the semismooth Newton method.