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：本文考虑标量半线性椭圆型偏微分方程，其非线性项为强单调但仅具有局部Lipschitz连续性。为线性化离散后的非线性问题，我们采用阻尼Zarantonello迭代，该方法将问题转化为对称正定的线性Poisson型方程。通过使用如局部光滑多重网格法等压缩代数求解器求解所得线性系统。我们构建了完全自适应算法，该算法能够均衡网格细化、迭代线性化及代数求解器产生的各类误差分量。理论证明表明：所提出的自适应迭代线性化有限元方法（AILFEM）可保证具有最优复杂度的收敛性，其中收敛阶基于整体计算代价（即计算时间）进行定义。数值实验探究了相关自适应参数的影响。