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arXiv · 2509.01531

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

Philipp Bringmann·Dirk Praetorius
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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.

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Philipp Bringmann, Dirk Praetorius. 2025-09-01. Global convergence of adaptive least-squares finite element methods for nonlinear PDEs. https://arxiv.org/abs/2509.01531

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