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

TL;DR

The paper develops an adaptive weighted least-squares finite element framework for nonlinear PDEs using the Zarantonello fixed-point iteration. By reformulating the problem as a nonlinear first-order system and solving each linearized step via a weighted LS functional, the authors obtain a built-in a posteriori error estimator that drives an adaptive Uzawa-type algorithm. They prove global R-linear convergence under a 2-growth condition in divergence form and analyze multiple weightings, identifying emphasized-gradient weighting as the most robust. The approach yields a convergent, mesh-adaptive scheme suitable for convex energy minimization and porous-media flow, with numerical experiments illustrating robustness and effectiveness in balancing discretization and linearization errors. Overall, the work provides a theoretically grounded, practically implementable path for reliable adaptive solution of nonlinear PDEs via LS FEM without requiring a Newton-type linearization upfront.

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.

Figures (8)

• Figure 1: Convergence and mesh plot as well as discrete solutions on the final mesh $\mathcal{T}^{40}_1$ (with $\#\mathcal{T}^{40}_1 = 548\,798$) for Algorithm \ref{['alg:adaptive']} applied to the convex energy minimization problem from Subsection \ref{['sec:convex_energy_minimization']}. The chosen parameters read $\delta = 1$, $\gamma = 0.9$, and $\theta = 0.3$. (Figure (\ref{['fig:nonlinear:flux']}) was created using the MATLAB function quiver2.mquiver2.)
• Figure 2: Convergence history plot of the error estimators \ref{['eq:applications:errors']} for Algorithm \ref{['alg:adaptive']} applied to the convex energy minimization problem from Subsection \ref{['sec:convex_energy_minimization']} with various reduction parameters $0 < \gamma < 1$. The remaining parameters read $\delta \in \{0.5, 1\}$ and $\theta = 0.3$.
• Figure 3: Convergence history plot of the estimators \ref{['eq:applications:errors']} for Algorithm \ref{['alg:adaptive']} applied to the convex energy minimization problem from Subsection \ref{['sec:convex_energy_minimization']} with various choices of the bulk parameter $0 < \theta \leq 1$. The remaining parameters read $\delta = 1$ and $\gamma = 0.9$.
• Figure 4: Convergence history plot for Algorithm \ref{['alg:adaptive']} applied to the convex energy minimization problem from Subsection \ref{['sec:convex_energy_minimization']} and various choices of the damping parameter $0 < \delta \leq 1$. The remaining parameters read $\theta = 0.3$ and $\gamma = 0.9$.
• Figure 5: Convergence history and condition number plot for Algorithm \ref{['alg:adaptive']} applied to the convex energy minimization problem from Subsection \ref{['sec:convex_energy_minimization']} with the weightings from Sections \ref{['sec:Zarantonello_least_squares']} and \ref{['sec:alternative_weightings']}. The chosen parameters read $\delta = 1$, $\theta = 0.3$, and $\gamma = 0.9$.

Theorems & Definitions (27)

• Remark 1: other discretizations
• Theorem 2: fundamental equivalence
• proof
• Theorem 3: plain convergence
• Lemma 4: nonlinear fundamental equivalence
• proof
• Proposition 5: a posteriori error estimates
• Lemma 6
• proof
• Theorem 7: well-posedness