Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Unconditional full linear convergence and optimal complexity of adaptive iteratively linearized FEM for nonlinear PDEs

Ani Miraçi, Dirk Praetorius, Julian Streitberger

TL;DR

The paper tackles nonlinear elliptic PDEs with strongly monotone operators by developing AILFEM, which tightly couples discretization, iterative linearization, and algebraic solving through a parameter-free stopping rule. The approach guarantees unconditional full R-linear convergence and, for suitably small adaptivity parameters, optimal complexity in the nonlinear approximation sense. Central to the contribution are the energy-based contraction framework, a uniformly bounded algebraic solver bound, and estimator-based adaptivity with Doerfler marking. The results offer parameter-robust convergence and practical efficiency for adaptive nonlinear FEM, with numerical experiments on L-shaped and Z-shaped domains validating the theory and demonstrating strong performance of the novel stopping criterion.

Abstract

We propose an adaptive iteratively linearized finite element method (AILFEM) in the context of strongly monotone nonlinear operators in Hilbert spaces. The approach combines adaptive mesh-refinement with an energy-contractive linearization scheme (e.g., the Kačanov method) and a norm-contractive algebraic solver (e.g., an optimal geometric multigrid method). Crucially, a novel parameter-free algebraic stopping criterion is designed and we prove that it leads to a uniformly bounded number of algebraic solver steps. Unlike available results requiring sufficiently small adaptivity parameters to ensure even plain convergence, the new AILFEM algorithm guarantees full R-linear convergence for arbitrary adaptivity parameters. Thus, parameter-robust convergence is guaranteed. Moreover, for sufficiently small adaptivity parameters, the new adaptive algorithm guarantees optimal complexity, i.e., optimal convergence rates with respect to the overall computational cost and, hence, time.

Unconditional full linear convergence and optimal complexity of adaptive iteratively linearized FEM for nonlinear PDEs

TL;DR

The paper tackles nonlinear elliptic PDEs with strongly monotone operators by developing AILFEM, which tightly couples discretization, iterative linearization, and algebraic solving through a parameter-free stopping rule. The approach guarantees unconditional full R-linear convergence and, for suitably small adaptivity parameters, optimal complexity in the nonlinear approximation sense. Central to the contribution are the energy-based contraction framework, a uniformly bounded algebraic solver bound, and estimator-based adaptivity with Doerfler marking. The results offer parameter-robust convergence and practical efficiency for adaptive nonlinear FEM, with numerical experiments on L-shaped and Z-shaped domains validating the theory and demonstrating strong performance of the novel stopping criterion.

Abstract

We propose an adaptive iteratively linearized finite element method (AILFEM) in the context of strongly monotone nonlinear operators in Hilbert spaces. The approach combines adaptive mesh-refinement with an energy-contractive linearization scheme (e.g., the Kačanov method) and a norm-contractive algebraic solver (e.g., an optimal geometric multigrid method). Crucially, a novel parameter-free algebraic stopping criterion is designed and we prove that it leads to a uniformly bounded number of algebraic solver steps. Unlike available results requiring sufficiently small adaptivity parameters to ensure even plain convergence, the new AILFEM algorithm guarantees full R-linear convergence for arbitrary adaptivity parameters. Thus, parameter-robust convergence is guaranteed. Moreover, for sufficiently small adaptivity parameters, the new adaptive algorithm guarantees optimal complexity, i.e., optimal convergence rates with respect to the overall computational cost and, hence, time.
Paper Structure (24 sections, 10 theorems, 109 equations, 7 figures, 2 tables, 1 algorithm)

This paper contains 24 sections, 10 theorems, 109 equations, 7 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Setting
  3. Weak formulation and inherent energy structure
  4. Discretization
  5. Iterative linearization
  6. Zarantonello (or Picard) linearization.
  7. Kačanov linearization.
  8. Damped Newton method.
  9. Algebraic solver
  10. Residual a-posteriori error estimator
  11. Comments
  12. Adaptive algorithm
  13. Index set $\mathcal{Q}$ for the triple loop
  14. Termination of algebraic solver and a posteriori control
  15. Full linear convergence
  16. ...and 9 more sections

Key Result

Proposition 1

Consider the model problem eq:nonlinear:strongform with the scalar nonlinearity $\boldsymbol{A}(\nabla u) = \mu(| \nabla u|^2) \nabla u$. Under the assumption eq:assumption-mu and for $p=1$, there exist constants $C_{\textnormal{stab}}, C_{\textnormal{rel}}, C_{\textnormal{drel}}, C_{\textnormal{mon The constant $C_{\textnormal{rel}}$ depends only on the uniform shape regularity of all meshes $\ma

Figures (7)

  • Figure 1: Adaptive mesh for the L-shaped domain problem from Section \ref{['section:HPW_benchmark']} with $\# \mathcal{T}_{7} = 6484$ (left) and for the Z-shaped domain problem from Section \ref{['section:GHPS_benchmark']} with $\# \mathcal{T}_{6} = 6122$ (right) for adaptivity parameters $\theta = 0.5$ and $\lambda_{\rm lin} = 0.7$.
  • Figure 2: Convergence of the error estimator $\eta_\ell(u_\ell^{{\underline{k}}, {\underline j}})$ and energy difference ${\rm d\!l}^2(u_\ell^{\underline{k}, \underline{j}}, u_\ell^{\underline{k}-1, \underline{j}})^{1/2}$ for the L-shaped domain problem from Section \ref{['section:HPW_benchmark']} with parameters $\theta = 0.5$ and $\lambda_{\rm lin} = 0.7$ as well as $\delta = 1/6$ (Zarantonello iteration) resp. $\delta = 0.5$ (Newton iteration).
  • Figure 3: Convergence of the error estimator $\eta_\ell(u_\ell^{{\underline{k}}, {\underline j}})$ and energy difference ${\rm d\!l}^2(u_\ell^{\underline{k}, \underline{j}}, u_\ell^{\underline{k}-1, \underline{j}})^{1/2}$ for the Z-shaped domain problem from Section \ref{['section:GHPS_benchmark']} with parameters $\theta = 0.5$ and $\lambda_{\rm lin} = 0.7$ as well as $\delta \approx 0.648364$ (Zarantonello iteration) resp. $\delta = 0.5$ (Newton iteration).
  • Figure 4: Convergence of the estimator $\eta_\ell(u_\ell^{{\underline{k}}, {\underline j}})$ and energy difference ${\rm d\!l}^2(u_\ell^{\underline{k}, \underline{j}}, u_\ell^{\underline{k}-1, \underline{j}})^{1/2}$ using the Kačanov linearization for the L-shaped domain problem from Section \ref{['section:HPW_benchmark']} (left) and Z-shaped domain problem from Section \ref{['section:GHPS_benchmark']} (right) for several polynomial degrees $p=1,2,3$.
  • Figure 5: History plot of $\alpha_\ell^{{\underline{k}}, {\underline j}}$ and $\alpha_{\min}$ (left), and iteration counters $\underline{j}[\ell, {\underline{k}}]$ and $J_{\max}$ (right) for the L-shaped domain problem from Section \ref{['section:HPW_benchmark']} with parameters $\theta = 0.5$ and $\lambda_{\rm lin} = 0.7$.
  • ...and 2 more figures

Theorems & Definitions (18)

  • Proposition 1: axioms of adaptivity
  • Lemma 2
  • Proposition 3: uniform bound on number of algebraic solvers steps
  • proof
  • Proposition 4: a posteriori error control
  • Theorem 5: full R-linear convergence of Algorithm \ref{['algorithm']}
  • Proposition 6: energy-contraction of inexact linearization
  • proof
  • proof : Proof of Theorem \ref{['theorem:linearconv']}
  • Corollary 7: rates = complexity bfmps2025
  • ...and 8 more