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

Cost-optimal adaptive FEM with linearization and algebraic solver for semilinear elliptic PDEs