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 mesh-refinement with an energy-contractive linearization scheme (e.g., the Ka\v{c}anov 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, parameter-robust 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.

翻译：我们提出了一种自适应迭代线性化有限元方法（AILFEM），适用于Hilbert空间中强单调非线性算子。该方法结合了自适应网格细化与能量收缩线性化方案（例如Kačanov方法）以及范数收缩代数求解器（例如最优几何多重网格方法）。关键创新在于设计了一种无参数代数终止准则，并证明该准则能够实现代数求解步骤数的均匀有界性。不同于现有结果需依赖充分小的自适应参数来保证基本收敛性，新AILFEM算法对任意自适应参数均能保证全R线性收敛性，从而确保参数鲁棒收敛性。此外，当自适应参数充分小时，该新自适应算法能保证最优复杂度，即关于整体计算成本（进而关于时间）的最优收敛率。