Global Convergence of Adaptive Least-Squares Finite Element Methods for Nonlinear Pdes

Abstract

The Zarantonello fixed-point iteration is an established linearization scheme for quasi-linear PDEs with strongly monotone and Lipschitz continuous nonlinearity in Hilbert spaces. This paper presents a weighted least-squares minimization for the computation of the update of this scheme. The resulting formulation allows for a conforming least-squares finite element discretization of the primal and dual variables of the PDE with arbitrary polynomial degree. The least-squares functional provides a built-in a posteriori discretization error estimator in each linearization step motivating an adaptive Uzawa-type algorithm with an outer linearization loop and an inner adaptive mesh-refinement loop. For quasi-linear PDEs in divergence form satisfying a 2-growth condition, we prove global R-linear convergence of the computed linearization iterates for arbitrary initial guesses. Particular focus is on the role of the weights in the least-squares functional of the linearized problem and their influence on the robustness of the Zarantonello damping parameter. Numerical experiments illustrate the performance of the proposed algorithm.