Ari Rappaport - Adaptive regularization, discretization, and regularization for nonsmooth elliptic PDE

Nous aurons l'occasion d'écouter Ari Rappaport (POEMS, ENSTA Paris) le 06 février 2025 en Amphi 2329 à l'ENSTA Paris.

 Adaptive regularization, discretization, and regularization for nonsmooth elliptic PDE 

We consider nonsmooth partial differential equations associated to 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. Then the finite element method is applied. We focus on the choice of the regularization parameter and adjust it on the basis of an 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 regularization, linearization, and discretization. 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.