Adaptive Inexact Semismooth Newton Methods for the Contact Problem Between Two Membranes

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 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 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, Ω) and on potential reconstructions in discrete subspaces of H 1 (Ω) 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.

A posteriori error estimates for a compositional two-phase flow with nonlinear complementarity constraints

“…At each time step n, (28) gives will lead to a system of 2N sp nonlinear equations. As we have 3N sp unknowns, to close the system, we use the nonlinear complementarity conditions as follows.…”

“…For a direct application of the min function see [14,28] and for more general details on C-functions see [39,40]. For 1 ≤ n ≤ N t let C n be any C-function satisfying…”

“…The mathematical models describing these complex phenomena are part of the large category of strongly nonlinear evolutive multiphase multi-compositional equations where numerical simulation appears to be the only viable approach to finding a solution. A key point investigated today is the reduction of the computational cost of the numerical resolution employing an adaptive strategy based on a posteriori error estimates [28,31,32,34,74].…”

“…Concerning a posteriori error estimate for variational inequalities, one can point out the pioneering work of Kornhuber [50], Chen and Nochetto [24], Veeser [72], Repin [67] and Ben Belgacem et al [10]. In particular in [10], a posteriori error estimates are given for exact solvers and recently, in [28] a posteriori error estimates are derived for inexact semismooth solvers and provide adaptive stopping criteria. The concept of adaptive stopping criteria relies on stopping the nonlinear and linear iterations whenever the associated estimators do not affect significantly the overall error, see [7,28,34,46,56].…”

“…In particular in [10], a posteriori error estimates are given for exact solvers and recently, in [28] a posteriori error estimates are derived for inexact semismooth solvers and provide adaptive stopping criteria. The concept of adaptive stopping criteria relies on stopping the nonlinear and linear iterations whenever the associated estimators do not affect significantly the overall error, see [7,28,34,46,56]. For multiphase flows, devising a posteriori error estimates between the exact solution and approximate solution seems very ambitious and is still an open problem.…”