On full linear convergence and optimal complexity of adaptive FEM with inexact solver

TL;DR

The paper addresses adaptive finite element methods with inexact solvers for broad classes of PDEs, introducing a novel tail-summability-based proof that achieves full R-linear convergence of a quasi-error without relying on a Pythagorean identity. By replacing the classical energy-orthogonality with a generalized quasi-orthogonality and leveraging axioms of adaptivity, it extends optimal complexity results to nonsymmetric and nonlinear, strongly monotone problems through single- and nested-solver AFEM frameworks. It proves that, under suitably small adaptivity and solver parameters, the convergence rates with respect to the number of degrees of freedom and the total computational work coincide, enabling practical efficiency gains. Numerical experiments corroborate the theory, showing parameter-tolerant, hp-robust performance across symmetric, nonsymmetric, and nonlinear PDEs, including cases with strong coefficient jumps and nonlinearities.

Abstract

The ultimate goal of any numerical scheme for partial differential equations (PDEs) is to compute an approximation of user-prescribed accuracy at quasi-minimal computational time. To this end, algorithmically, the standard adaptive finite element method (AFEM) integrates an inexact solver and nested iterations with discerning stopping criteria balancing the different error components. The analysis ensuring optimal convergence order of AFEM with respect to the overall computational cost critically hinges on the concept of R-linear convergence of a suitable quasi-error quantity. This work tackles several shortcomings of previous approaches by introducing a new proof strategy. First, the algorithm requires several fine-tuned parameters in order to make the underlying analysis work. A redesign of the standard line of reasoning and the introduction of a summability criterion for R-linear convergence allows us to remove restrictions on those parameters. Second, the usual assumption of a (quasi-)Pythagorean identity is replaced by the generalized notion of quasi-orthogonality from [Feischl, Math. Comp., 91 (2022)]. Importantly, this paves the way towards extending the analysis to general inf-sup stable problems beyond the energy minimization setting. Numerical experiments investigate the choice of the adaptivity parameters.

Figures (13)

• Figure 1: Illustration of the initial triangulation $\mathcal{T}_{0}$, the adaptively generated mesh $\mathcal{T}_{15}$ with $518$ triangles, the exact solution $u^\star$ and the computed solution $u_{15}^{\underline{k}}$ for the Kellogg benchmark problem \ref{['eq:experiment:symmetric']} with polynomial degree $p = 2$, marking parameter $\theta = 0.5$, and algebraic solver parameter $\lambda = 0.01$.
• Figure 2: Convergence history plot of the error estimator $\eta_{\ell}(u_{\ell}^{\underline{k}})$ with respect to the number of degrees of freedom (left) and the cumulative computation time (right) for the Kellogg benchmark problem \ref{['eq:experiment:symmetric']} for different polynomial degrees $p \in \{1, 2, 3, 4\}$ with fixed marking parameter $\theta = 0.5$ and algebraic solver parameter $\lambda = 0.01$.
• Figure 3: Convergence history plot of the error estimator $\eta_{\ell}(u_{\ell}^{\underline{k}})$ for different algebraic solver parameters $\lambda \in \{0.001, 0.01, 0.1, 0.5, 1\}$ and fixed polynomial degree $p=2$ and marking parameter $\theta = 0.5$ with respect to the number of degrees of freedom (left) and the cumulative computation time (right) for the Kellogg benchmark problem \ref{['eq:experiment:symmetric']}.
• Figure 4: Comparison of the cumulative computation time for the algebraic solver (Algorithm \ref{['algorithm:single']}) from imps2022 with the Matlab built-in mldivide (Algorithm \ref{['algorithm:exact']}) over the cumulative number of degrees of freedom to solve the Kellogg benchmark problem \ref{['eq:experiment:symmetric']} with polynomial degree $p \in \{1, 4\}$, marking parameter $\theta = 0.5$, and algebraic solver parameter $\lambda = 0.01$.
• Figure 5: Illustration of the initial triangulation $\mathcal{T}_{0}$ and the sequence of adaptively generated meshes $\mathcal{T}_{0}, \ldots, \mathcal{T}_{4}$ for the experiment \ref{['eq:experiment']}.

Theorems & Definitions (30)

• Remark 1
• Proposition 2: axioms of adaptivity
• Proposition 3: validity of quasi-orthogonality
• Remark 4
• Theorem 5: R-linear convergence of Algorithm \ref{['algorithm:exact']}
• Remark 6
• Lemma 7: tail summability criterion
• proof : Proof of Theorem \ref{['theorem:exact:convergence']}
• Theorem 8: full R-linear convergence of Algorithm \ref{['algorithm:single']}
• Remark 9