Adaptive Iterative Linearization Galerkin Methods for Nonlinear Problems

Pascal Heid, Thomas P. Wihler

Published 2018-08-15Version 1

Fixed point iterations are widely used for the analysis and numerical treatment of nonlinear problems. In many cases, such schemes can be interpreted as iterative local linearization methods, which, as will be shown, can be obtained by applying a suitable preconditioning operator to the original (nonlinear) equation. Based on this observation, we will derive a unified abstract framework which recovers some prominent iterative schemes. In particular, for Lipschitz continuous and strongly monotone operators, we derive a general convergence analysis. Furthermore, in order to solve nonlinear problems numerically, we propose a combination of the iterative linearization approach and the classical Galerkin discretization method, thereby giving rise to the so-called iterative linearization Galerkin (ILG) methodology. Moreover, still on an abstract level, based on two different elliptic reconstruction techniques, we derive a posteriori error estimates which separately take into account the discretization and linearization errors. Furthermore, we propose an adaptive algorithm, which provides an efficient interplay between these two effects. Finally, our abstract theory and the performance of the adaptive ILG approach are illustrated by means of a finite element discretization scheme for various examples of quasilinear stationary conservation laws.