Adaptive inexact semismooth Newton methods for the contact problem between two membranes

Jad Dabaghi (1, 2) , Vincent Martin (3, 4) , Martin Vohralík (1, 2)

1 SERENA - Simulation for the Environment: Reliable and Efficient Numerical Algorithms
2 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
3 LMAC - Laboratoire de Mathématiques Appliquées de Compiègne
4 UTC - Université de Technologie de Compiègne

We propose an adaptive inexact version of a class of semismooth Newton methods. As a model problem, we study the system of variational inequalities describing the contact between two membranes. This problem is discretized with conforming finite elements of order $p \geq 1$, yielding a nonlinear algebraic system of variational inequalities. We consider any iterative semismooth linearization algorithm like the Newton-min or the Newton--Fischer--Burmeister which we complement by any iterative linear algebraic solver. We then derive an a posteriori estimate on the error between the exact solution and the approximate solution which is valid at any step of the linearization and algebraic resolutions. Our estimate is based on flux reconstructions in discrete subspaces of $\mathbf{H}(\mathrm{div}, \Omega)$ and on potential reconstructions in discrete subspaces of $H^1(\Omega)$ satisfying the constraints. It distinguishes the discretization, linearization, and algebraic components of the error. Consequently, we can formulate adaptive stopping criteria for both solvers, giving rise to an adaptive version of the considered inexact semismooth Newton algorithm. Under these criteria, the efficiency of our estimates is also established, meaning that we prove them equivalent with the error up to a generic constant, except for a typically small contact term. Numerical experiments for the Newton-min algorithm in combination with the GMRES algebraic solver confirm the efficiency of the developed adaptive method.

Mots clés: Variational inequality; Adaptivity; Semismooth Newton method; A posteriori error estimate; Contact problem; Stopping criterion; Complementarity condition

Domaines: Analyse numérique [math.NA]

Jad Dabaghi, Vincent Martin, Martin Vohralík. Adaptive inexact semismooth Newton methods for the contact problem between two membranes. 2018. ⟨hal-01666845v2⟩