Adaptive Iterative Linearization Galerkin Methods for Nonlinear Problems

TL;DR

This paper introduces an adaptive iterative linearization Galerkin framework for solving nonlinear equations, combining linearization and discretization, with error estimates and an adaptive algorithm, demonstrated on finite element discretizations of quasilinear elliptic PDEs.

Contribution

It develops a unified abstract framework for iterative linearization methods, integrates Galerkin discretization, and proposes an adaptive algorithm with error control for nonlinear PDEs.

Findings

Convergence analysis for Lipschitz continuous and strongly monotone operators.

Development of a posteriori error estimates separating discretization and linearization errors.

Numerical experiments demonstrating the efficiency of the ILG method.

Abstract

A wide variety of (fixed-point) iterative methods for the solution of nonlinear equations (in Hilbert spaces) exists. 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 the context of numerical solution schemes for nonlinear partial differential equations, 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. In addition, the ILG approach will be applied to the specific context of finite element discretizations of quasilinear elliptic equations, and some numerical experiments will be performed.