Adaptive regularization, discretization, and linearization for nonsmooth problems based on primal-dual gap estimators

We consider nonsmooth partial differential equations associated with a minimization of an energy functional. We adaptively regularize the nonsmooth nonlinearity so as to be able to apply the usual Newton linearization, which is not always possible otherwise. We apply the finite element method as a discretization. We focus on the choice of the regularization parameter and adjust it on the basis of an a posteriori error estimate for the difference of energies of the exact and approximate solutions. Importantly, our estimates distinguish the different error components, namely those of regularization, linearization, and discretization. This leads to an algorithm that steers the overall procedure by adaptive stopping criteria with parameters for the regularization, linearization, and discretization levels. We prove guaranteed upper bounds for the energy difference and discuss the robustness of the estimates with respect to the magnitude of the nonlinearity when the stopping criteria are satisfied. Numerical results illustrate the theoretical developments.