Séance Séminaire

Séminaire ACSIOM

Adaptive inexact Newton methods and adaptive regularization and space--time discretization for unsteady nonlinear problems

(collaboration avec A. Ern, D. Di Pietro, and S. Yousef) We first describe the adaptive inexact Newton method of the reference [1]. Herein, to solve a nonlinear algebraic system arising from a numerical discretization of a steady nonlinear partial differential equation, we consider an iterative linearization (for example the Newton or the fixed-point ones), and, on its each step, an iterative algebraic solver (for example the conjugate gradients or GMRes). We derive adaptive stopping criteria for both these iterative solvers. Our criteria are based on an a posteriori error estimate which distinguishes the different error components, namely the discretization error, the linearization error, and the algebraic error. We stop the iterations whenever the corresponding error does no longer affect the overall error significantly. Our estimates yield a guaranteed upper bound on the overall error , as well as (local) efficiency and robustness with respect to the size of the nonlinearity. We then turn to a model nonlinear unsteady (degenerate) problem, the Stefan solidification, and consider its conforming spatial and backward Euler temporal discretizations. Following [2], we show how to derive estimators yielding a guaranteed and fully computable upper bound on the dual norm of the residual, as well as on the L^2(L^2) error of the temperature and the L^2(H^(-1)) error of the enthalpy. Extending the above steady case developments, we also distinguish all the error components and propose an adaptive algorithm. A theoretical proof of the efficiency of our estimate, as well as illustrative numerical experiments, are presented. [1] A. Ern and M. Vohralik, Adaptive inexact Newton methods with a posteriori stopping criteria for nonlinear diffusion PDEs, SIAM J. Sci. Comput. 35 (2013), A1761-A1791. [2] D. Di Pietro, M. Vohralik, and S. Yousef, Adaptive regularization, linearization, and discretization and a posteriori error control for the two-phase Stefan problem, Math. Comp. 84 (2015), 153-186.