Convergence and quasi-optimal cost of adaptive algorithms for nonlinear operators including iterative linearization and algebraic solver

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