Rate optimal adaptive FEM with inexact solver for nonlinear operators. (15th September 2017)

• Record Type: Journal Article
• Authors: Gantner, Gregor; Haberl, Alexander; Praetorius, Dirk; Stiftner, Bernhard
• Is Part Of: IMA journal of numerical analysis. Volume 38:Number 4(2018)
• Page Start: 1797
• Page End: 1831
• Publication Date: 2017-09-15
• Subjects: quasilinear elliptic PDE -- finite element method -- adaptive mesh-refinement -- adaptive solution of nonlinear algebraic system -- optimal convergence rates -- Banach fixed point theorem; Numerical analysis -- Periodicals; 519.405
• DOI: 10.1093/imanum/drx050
• Languages: English
• ISSNs: 0272-4979

Abstract

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 Picard iterations is uniformly bounded in generic cases, and the overall computational cost is (almost) optimal. Numerical experiments confirm the theoretical results.