Parameterrobust full linear convergence and optimal complexity of adaptive iteratively linearized FEM for nonlinear PDEs
Abstract
We propose an adaptive iteratively linearized finite element method (AILFEM) in the context of strongly monotone nonlinear operators in Hilbert spaces. The approach combines adaptive meshrefinement with an energycontractive linearization scheme (e.g., the Kačanov method) and a normcontractive algebraic solver (e.g., an optimal geometric multigrid method). Crucially, a novel parameterfree 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 Rlinear convergence for arbitrary adaptivity parameters. Thus, parameterrobust convergence is guaranteed. Moreover, for sufficiently small adaptivity parameters, the new adaptive algorithm guarantees optimal complexity, i.e., optimal convergence rates with respect to the overall computational cost and, hence, time.
 Publication:

arXiv eprints
 Pub Date:
 January 2024
 DOI:
 10.48550/arXiv.2401.17778
 arXiv:
 arXiv:2401.17778
 Bibcode:
 2024arXiv240117778M
 Keywords:

 Mathematics  Numerical Analysis