We consider a second-order elliptic boundary value problem with strongly monotone and Lipschitz-continuous nonlinearity. We design and study its adaptive numerical approximation interconnecting a finite element discretization, the Banach–Picard linearization, and a contractive linear algebraic solver. In particular, we 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) Banach–Picard linearization to remain contractive. Similarly, we identify suitable stopping criteria for the Banach–Picard iteration that leave an amount of linearization error that is not harmful for the residual a posteriori error estimate to steer reliably the adaptive mesh-refinement. For the resulting algorithm, we prove a contraction of the (doubly) inexact iterates after some amount of steps of mesh-refinement/linearization/algebraic solver, leading to its linear convergence. Moreover, for usual mesh-refinement 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 mesh-refinement, 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.

我们考虑具有强单调和Lipschitz连续非线性的二阶椭圆形边值问题。我们设计并研究了它的自适应数值逼近，它将有限元离散化，Banach-Picard线性化和收缩线性代数求解器联系起来。特别是，我们确定了代数求解器的停止准则，该准则一方面不要求过紧的公差，但另一方面足以使不精确的（扰动的）Banach-Picard线性化保持收缩。同样地，我们为Banach-Picard迭代确定合适的停止准则，该准则保留了一些线性化误差，该误差对剩余的后验无害误差估计以可靠地引导自适应网格细化。对于生成的算法，我们证明了经过一定数量的网格细化/线性化/代数求解器步骤后，（双）不精确迭代的收缩，从而导致其线性收敛。此外，对于通常的网格细化规则，我们还证明，相对于相对于初始网格添加的元素数量（自由度），总误差以最佳速率衰减。最终，我们证明了我们的完全自适应算法以相同的最佳速率降低了总体误差，并且相对于总体算法成本而言，总体算法成本表示为所有网格细化，线性化和代数求解器步骤中网格元素数量的总和。