Global convergence of adaptive least-squares finite element methods for nonlinear PDEs

Abstract

The Zarantonello fixed-point iteration is an established linearization scheme for quasilinear 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 variable 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 quasilinear 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.

arXiv commentNumerical results can be reproduced using the MATLAB software package on Code Ocean: https://doi.org/10.24433/CO.0796231.v1