Newton's method in adaptive iteratively linearized FEM

TL;DR

This paper integrates Newton's method into an adaptive finite element framework for nonlinear PDEs, introducing a novel residual-based convergence analysis that achieves optimal rates beyond energy-minimization problems.

Contribution

It presents the first convergence analysis with optimal rates for an adaptive iteratively linearized FEM using a residual-based measure, applicable to strongly monotone operators.

Findings

Achieves global linear and local quadratic convergence.

Provides numerical evidence supporting theoretical convergence rates.

Extends adaptive FEM analysis to a broader class of nonlinear PDEs.

Abstract

This paper concerns the inclusion of Newton's method into an adaptive finite element method (FEM) for the solution of nonlinear partial differential equations (PDEs). It features an adaptive choice of the damping parameter in the Newton iteration for the discretized nonlinear problems on each level ensuring both global linear and local quadratic convergence. In contrast to energy-based arguments in the literature, a novel approach in the analysis considers the discrete dual norm of the residual as a computable measure for the linearization error. As a consequence, this paper provides the first convergence analysis with optimal rates of an adaptive iteratively linearized FEM beyond energy-minimization problems. The presented theory applies to strongly monotone operators with locally Lipschitz continuous Fr\'echet derivative. We present a class of semilinear PDEs fitting into this framework and provide numerical experiments to underline the theoretical results.