Adaptive Inexact Semismooth Newton Methods for the Contact Problem Between Two Membranes
- PDF / 2,266,169 Bytes
- 32 Pages / 439.37 x 666.142 pts Page_size
- 89 Downloads / 187 Views
Adaptive Inexact Semismooth Newton Methods for the Contact Problem Between Two Membranes Jad Dabaghi1,2
· Vincent Martin3 · Martin Vohralík1,2
Received: 19 October 2018 / Revised: 14 April 2020 / Accepted: 14 June 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We propose an adaptive inexact version of a class of semismooth Newton methods that is aware of the continuous (variational) level. 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 ≥ 1, yielding a nonlinear algebraic system with 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 at the continuous level 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 H(div, ) and on potential reconstructions in discrete subspaces of H 1 () 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 the leading estimates is also established, meaning that we prove them equivalent with the error up to a generic constant. Numerical experiments for the Newton-min algorithm in combination with the GMRES algebraic solver confirm the efficiency of the developed adaptive method.
This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant Agreement No. 647134 GATIPOR).
B
Jad Dabaghi [email protected] Vincent Martin [email protected] Martin Vohralík [email protected]
1
Inria, 2 rue Simone Iff, 75589 Paris, France
2
Université Paris-Est, CERMICS (ENPC), 77455 Marne-la-Vallée 2, France
3
Université de technologie de Compiègne (UTC), LMAC (Laboratory of Applied Mathematics of Compiègne), CS 60319, 60203 Compiègne Cedex, France 0123456789().: V,-vol
123
28
Page 2 of 32
Journal of Scientific Computing
(2020) 84:28
Keywords Variational inequality · Complementarity condition · Contact problem · Semismooth Newton method · A posteriori error estimate · Adaptivity · Stopping criterion
1 Introduction Consider a system of algebraic inequalities written in the following form: find a vector X h ∈ Rn , such that EX h = F, K (X h ) ≥ 0, G(X h ) ≥ 0,
K (X h ) · G(X h ) = 0,
(1)
where, for some integers n > 1 and 0 < m < n, E ∈ Rn−m,n is a matrix, K : Rn → Rm and G : Rn → Rm are affine operators, and F ∈ Rn−m is a given vector. The first line of (1) typically represents the discretization of a linear partia
Data Loading...
