Cost-optimal adaptive FEM with linearization and algebraic solver for semilinear elliptic PDEs

TL;DR

This work develops an adaptive iteratively linearized finite element method (AILFEM) for semilinear elliptic PDEs with strongly monotone, locally Lipschitz nonlinearities, casting the problem as a strongly monotone operator equation and solving it with a damped Zarantonello linearization inside a fully adaptive loop. By coordinating mesh refinement (via NVB), linearization, and an inexact contractive algebraic solver, the authors prove uniform boundedness of iterates, full $R$-linear convergence, and optimal complexity with respect to computational cost. The approach extends AFEM theory to locally Lipschitz nonlinearities and demonstrates that optimal rates, both in degrees of freedom and time, are achievable under reasonable adaptivity and solver stopping criteria. Numerical experiments with an hp-robust multigrid show robust performance across various adaptivity parameters and nonlinear regimes, supporting the theoretical convergence and complexity results. This provides a practical and scalable framework for efficiently solving nonlinear PDEs in computational mechanics where nonlinearity is monotone.

Abstract

We consider scalar semilinear elliptic PDEs, where the nonlinearity is strongly monotone, but only locally Lipschitz continuous. To linearize the arising discrete nonlinear problem, we employ a damped Zarantonello iteration, which leads to a linear Poisson-type equation that is symmetric and positive definite. The resulting system is solved by a contractive algebraic solver such as a multigrid method with local smoothing. We formulate a fully adaptive algorithm that equibalances the various error components coming from mesh refinement, iterative linearization, and algebraic solver. We prove that the proposed adaptive iteratively linearized finite element method (AILFEM) guarantees convergence with optimal complexity, where the rates are understood with respect to the overall computational cost (i.e., the computational time). Numerical experiments investigate the involved adaptivity parameters.

Figures (5)

• Figure 1: Depiction of the nested loops of the AILFEM algorithm \ref{['algorithm:AILFEM']} below.
• Figure 2: Experiment \ref{['example:gordon1']}: Convergence plots of the error $|\mkern-1.5mu|\mkern-1.5mu| u^\star-u_\ell^{\underline{k},\underline{i}} |\mkern-1.5mu|\mkern-1.5mu|$ (diamond) and the error estimator $\eta_\ell(u_\ell^{\underline{k}, \underline{i}})$ (circle) over $\mathtt{cost}(\ell, \underline{k}, \underline{i})$ (left) and over computational time in seconds (right).
• Figure 3: Experiment \ref{['experiment:quasilinear']}: Convergence plots of various error components over the degrees of freedom (left). Right: Plot of the approximate solution $u_{13}^{1, 1}$ on $\mathcal{X}_{13}$ with $\# \mathcal{X}_{13} =10209$.
• Figure 4: Convergence plots of the error estimator $\eta_\ell(u^{\underline{k}, \underline{i}}_\ell)$ over computational time of Experiment \ref{['example:gordon2']}. Left: Convergence plot for $p=1$ Right: Convergence plot for $p=3$.
• Figure 5: Mesh plot of Experiment \ref{['example:gordon2']} for $p=3$. Left: Adaptivity parameter $\lambda_{\textup{lin}} = 0.2$. Right: Adaptivity parameter $\lambda_{\textup{lin}} = 0.7$.

Theorems & Definitions (22)

• Remark 1
• Proposition 2: bbimp2022cost
• Lemma 3: see, e.g., ghps2018
• Proposition 4: see, e.g., bbimp2022cost
• Proposition 5: bbimp2022
• Lemma 6
• proof
• Theorem 7
• Lemma 8
• proof