The talk **Adaptive inexact Newton methods**, by Martin Vohralík (INRIA de Paris), will be held on Monday March 21, 2016 at 9:00 in room **K6**.

**Abstract**

We describe in this lecture an adaptive inexact Newton method. 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 hinge on equilibrated flux reconstructions. They yield a guaranteed upper bound on the overall error measured by the dual norm of the residual augmented by a jump nonconformity term. Our estimates are valid at each step of the nonlinear and linear solvers. Importantly, we prove their (local) efficiency and robustness with respect to the size of the nonlinearity. Details can be found in “Adaptive inexact Newton methods with a posteriori stopping criteria for nonlinear diffusion PDEs, *SIAM J. Sci. Comput.* **35** (2013), A1761–A1791.”