Work

Unconditional full linear convergence and optimal complexity of adaptive iteratively linearized FEM for nonlinear PDEs

Year: 2025

Type: article

Abstract: We propose an adaptive iteratively linearized finite element method (AILFEM) in the context of strongly monotone nonlinear operators with energy structure in Hilbert spaces. The approach combines adaptive mesh-refinement with an energy-contractive linearization scheme (e.g., the Kačanov method) and a norm-contractive algebraic solver (e.g., an optimal geometric multigrid method). Crucially, a novel parameter-free algebraic stopping criterion is designed and we prove that it leads to a uniformly bounded number of algebraic solver steps. Unlike available results requiring sufficiently small adaptivity parameters to ensure even plain convergence, the new AILFEM algorithm guarantees full R-linear convergence for arbitrary adaptivity parameters. Thus, unconditional convergence is guaranteed. Moreover, for sufficiently small adaptive mesh-refinement parameter θ \theta and linearization-stopping parameter λ l i n \lambda _{\mathrm {lin}} , the new adaptive algorithm guarantees optimal complexity, i.e., optimal convergence rates with respect to the overall computational cost and hence time.

Source: Mathematics of Computation

DOI: 10.1090/mcom/4135

Authors: Ani Miraçi, Dirk Praetorius, Julian Streitberger

Institutions: TU Wien, Sorbonne Université, Laboratoire Jacques-Louis Lions

Language: English

Topic: Advanced Numerical Methods in Computational Mathematics

Subfield: Computational Mechanics

Field: Engineering

Domain: Physical Sciences

Open Access status: green

Funder: Austrian Science Fund

Awards: G3878859087, Computational nonlinear PDEs, Functional error estimates for PDEs on unbounded domains, G9000005545, G9000174648