Convergence and quasioptimal cost of adaptive algorithms for nonlinear operators including iterative linearization and algebraic solver
Abstract
We consider a secondorder elliptic boundary value problem with strongly monotone and Lipschitzcontinuous nonlinearity. We design and study its adaptive numerical approximation interconnecting a finite element discretization, the BanachPicard linearization, and a contractive linear algebraic solver. We in particular identify stopping criteria for the algebraic solver that on the one hand do not request an overly tight tolerance but on the other hand are sufficient for the inexact (perturbed) BanachPicard linearization to remain contractive. Similarly, we identify suitable stopping criteria for the BanachPicard iteration that leave an amount of linearization error that is not harmful for the residual aposteriori error estimate to steer reliably the adaptive meshrefinement. For the resulting algorithm, we prove a contraction of the (doubly) inexact iterates after some amount of steps of meshrefinement / linerization / algebraic solver, leading to its linear convergence. Moreover, for usual meshrefinement rules, we also prove that the overall error decays at the optimal rate with respect to the number of elements (degrees of freedom) added with respect to the initial mesh. Finally, we prove that our fully adaptive algorithm drives the overall error down with the same optimal rate also with respect to the overall algorithmic cost expressed as the cumulated sum of the number of mesh elements over all meshrefinement, linearization, and algebraic solver steps. Numerical experiments support these theoretical findings and illustrate the optimal overall algorithmic cost of the fully adaptive algorithm on several test cases.
 Publication:

arXiv eprints
 Pub Date:
 April 2020
 DOI:
 10.48550/arXiv.2004.13137
 arXiv:
 arXiv:2004.13137
 Bibcode:
 2020arXiv200413137H
 Keywords:

 Mathematics  Numerical Analysis;
 65N12;
 65N15;
 65N30;
 65N50;
 68Q25
 EPrint:
 Numerische Mathematik (2021)