Rate optimal adaptive FEM with inexact solver for nonlinear operators

TL;DR

This paper establishes convergence and optimal computational efficiency for an adaptive finite element method solving nonlinear equations with strongly monotone operators, incorporating inexact Picard iterations and nested iteration techniques.

Contribution

It introduces a novel analysis that includes inexact nonlinear solves via Picard iteration, proving uniform bounds and near-optimal computational cost for adaptive FEM.

Findings

Convergence with optimal algebraic rates is proven.

The number of Picard iterations remains uniformly bounded.

Numerical experiments confirm theoretical predictions.

Abstract

We prove convergence with optimal algebraic rates for an adaptive finite element method for nonlinear equations with strongly monotone operator. Unlike prior works, our analysis also includes the iterative and inexact solution of the arising nonlinear systems by means of the Picard iteration. Using nested iteration, we prove, in particular, that the number of of Picard iterations is uniformly bounded in generic cases, and the overall computational cost is (almost) optimal. Numerical experiments confirm the theoretical results.