Convergence of an adaptive Ka\v{c}anov FEM for quasi-linear problems

TL;DR

This paper introduces an adaptive finite element method combining Kačanov iteration and mesh refinement for quasi-linear elliptic problems, proving convergence and demonstrating effectiveness through numerical experiments.

Contribution

It develops a convergent adaptive FEM algorithm based on inexact Kačanov iteration with mesh adaptation for quasi-linear problems.

Findings

The method converges for any reasonable marking strategy.

Convergence is proven from any initial mesh.

Numerical experiments confirm theoretical results.

Abstract

We design an adaptive finite element method to approximate the solutions of quasi-linear elliptic problems. The algorithm is based on a Ka\v{c}anov iteration and a mesh adaptation step is performed after each linear solve. The method is thus \emph{inexact} because we do not solve the discrete nonlinear problems exactly, but rather perform one iteration of a fixed point method (Ka\v{c}anov), using the approximation of the previous mesh as an initial guess. The convergence of the method is proved for any \emph{reasonable} marking strategy and starting from any initial mesh. We conclude with some numerical experiments that illustrate the theory.