We consider scalar semilinear elliptic PDEs where the nonlinearity is strongly monotone, but only locally Lipschitz continuous. We formulate an adaptive iterative linearized finite element method (AILFEM) which steers the local mesh refinement as well as the iterative linearization of the arising nonlinear discrete equations. To this end, we employ a damped Zarantonello iteration so that, in each step of the algorithm, only a linear Poisson-type equation has to be solved. We prove that the proposed AILFEM strategy guarantees convergence with optimal rates, where rates are understood with respect to the overall computational complexity (i.e., the computational time). Moreover, we formulate and test an adaptive algorithm where also the damping parameter of the Zarantonello iteration is adaptively adjusted. Numerical experiments underline the theoretical findings.

翻译：我们考虑标量半线性椭圆偏微分方程，其中非线性项是强单调的，但仅局部Lipschitz连续。我们构造了一种自适应迭代线性化有限元方法（AILFEM），该方法同时指导局部网格细化以及由此产生的非线性离散方程的迭代线性化。为此，我们采用阻尼Zarantonello迭代，使得在算法的每一步中，只需求解一个线性泊松型方程。我们证明了所提出的AILFEM策略能够保证以最优速率收敛，其中速率是相对于整体计算复杂度（即计算时间）而言的。此外，我们还构造并测试了一种自适应算法，其中Zarantonello迭代的阻尼参数也进行自适应调整。数值实验验证了理论结果。