ANIMALS: Adaptive multilevel inexact Newton methods with smoothing for semilinear problems

Cascadic algorithms solve a nonlinear problem on the coarsest level and then successively refine the mesh, applying a classical iterative scheme, called a smoother in this context, on each refinement level. Our Adaptive Inexact Newton Multilevel Algorithm (ANIMAL) has this structure, with an outer loop that mimics an inexact Newton iteration at the continuous level and an inner loop that allows multiple adaptive refinement steps. This distinguishes ANIMAL from traditional multilevel schemes and is motivated by the mismatch between the locally quadratic convergence of Newton’s method and the merely linear error reduction achieved by one-level mesh refinement. Classical adaptive mesh refinement, however, requires an accurate solve at every refinement step. We therefore introduce a less costly smoothing-enhanced variant, ANIMALS, which replaces accurate intermediate solves by a fixed number m of smoothing iterations. We establish R -quadratic convergence for the underlying inexact Newton scheme, while numerical experiments for semilinear elliptic, convection–diffusion, and Ginzburg–Landau problems demonstrate the effectiveness of the proposed approach.