We consider scalar semilinear elliptic PDEs, where the nonlinearity is strongly monotone, but only locally Lipschitz continuous. To linearize the arising discrete nonlinear problem, we employ a damped Zarantonello iteration, which leads to a linear Poisson-type equation that is symmetric and positive definite. The resulting system is solved by a contractive algebraic solver such as a multigrid method with local smoothing. We formulate a fully adaptive algorithm that equibalances the various error components coming from mesh refinement, iterative linearization, and algebraic solver. We prove that the proposed adaptive iteratively linearized finite element method (AILFEM) guarantees convergence with optimal complexity, where the rates are understood with respect to the overall computational cost (i.e., the computational time). Numerical experiments investigate the involved adaptivity parameters.

翻译：本文研究标量半线性椭圆偏微分方程，其非线性项具有强单调性，但仅满足局部Lipschitz连续性。为线性化离散非线性问题，我们采用带阻尼的Zarantonello迭代法，该方法将原问题转化为对称正定的线性泊松型方程。所得方程组通过收缩性代数求解器（如采用局部平滑的多重网格法）进行求解。我们构建了一种完全自适应算法，能够均衡来自网格细化、迭代线性化及代数求解器的各类误差分量。理论证明表明，所提出的自适应迭代线性化有限元法（AILFEM）在最优复杂度意义下保证收敛性，其中收敛速率是基于整体计算成本（即计算时间）进行衡量的。数值实验进一步探究了所涉及的自适应参数。