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Unconditional full linear convergence and quasi-optimal complexity of smoothed adaptive finite element methods

Philipp Bringmann, Christoph Lietz, Dirk Praetorius

Abstract

We present the first rigorous convergence analysis of the smoothed adaptive finite element method (S-AFEM) proposed in [Mulita, Giani, Heltai: SIAM J. Sci. Comput. 43, 2021]. S-AFEM modifies the classical adaptive finite element method (AFEM) by performing accurate discrete solves only on periodically determined mesh levels, while the intermediate levels employ a fixed number of cheap smoothing iterations. Numerical experiments in that work showed that this strategy generates adapted meshes comparable to those of AFEM at substantially lower computational cost. In this paper, we prove unconditional full R-linear convergence of a suitable quasi-error quantity and, for sufficiently small adaptivity parameters, optimal convergence rates with respect to the overall computational cost. The analysis requires only a mild uniform stability assumption on the employed smoother, satisfied by standard methods such as Richardson, Gauss-Seidel, conjugate gradient, and multigrid schemes. Our results apply to general second-order linear elliptic PDEs and show that S-AFEM retains all desired abstract convergence guarantees of AFEM while reducing the cumulative computational time. Numerical experiments validate the theory, analyze runtime performance, and underline the potential of S-AFEM for speed-up in AFEM computations.

Unconditional full linear convergence and quasi-optimal complexity of smoothed adaptive finite element methods

Abstract

We present the first rigorous convergence analysis of the smoothed adaptive finite element method (S-AFEM) proposed in [Mulita, Giani, Heltai: SIAM J. Sci. Comput. 43, 2021]. S-AFEM modifies the classical adaptive finite element method (AFEM) by performing accurate discrete solves only on periodically determined mesh levels, while the intermediate levels employ a fixed number of cheap smoothing iterations. Numerical experiments in that work showed that this strategy generates adapted meshes comparable to those of AFEM at substantially lower computational cost. In this paper, we prove unconditional full R-linear convergence of a suitable quasi-error quantity and, for sufficiently small adaptivity parameters, optimal convergence rates with respect to the overall computational cost. The analysis requires only a mild uniform stability assumption on the employed smoother, satisfied by standard methods such as Richardson, Gauss-Seidel, conjugate gradient, and multigrid schemes. Our results apply to general second-order linear elliptic PDEs and show that S-AFEM retains all desired abstract convergence guarantees of AFEM while reducing the cumulative computational time. Numerical experiments validate the theory, analyze runtime performance, and underline the potential of S-AFEM for speed-up in AFEM computations.
Paper Structure (21 sections, 10 theorems, 109 equations, 9 figures, 3 tables, 1 algorithm)

This paper contains 21 sections, 10 theorems, 109 equations, 9 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Model problem
  4. Triangulations and refinement
  5. Discretization
  6. Error estimator
  7. Discrete iterative smoothers
  8. Uniform stability and contraction
  9. Uniformly stable smoothers for symmetric PDEs
  10. Uniformly stable smoothers for non-symmetric PDEs
  11. Smoothed AFEM algorithm
  12. Full R-linear convergence
  13. Quasi-optimal complexity
  14. Numerical experiments
  15. Implementation
  16. ...and 6 more sections

Key Result

Proposition 1

There exist $C_{\textnormal{stab}}, C_{\textnormal{rel}}, C_{\textnormal{drel}}>0$, $0<q_{\textnormal{red}}<1$, and $C_{\textnormal{mon}}\geq 1$ such that, for any $\mathcal{T}_H\in\mathbb{T}$, any $\mathcal{T}_h\in\mathop{\mathrm{\mathtt{refine}}}\nolimits(\mathcal{T}_H)$, any $\mathcal{U}_H\subset The constant $C_{\textnormal{rel}}$ depends only on the dimension $d$ and uniform shape regularity,

Figures (9)

  • Figure 1: Refinement indicators for a Poisson problem for an initial approximation (left), three Gauss--Seidel iterations (center), and the exact discrete solution (right).
  • Figure 2: Illustration of adaptive refinement loops.
  • Figure 3: Final meshes $\mathcal{T}_{15}$ on level $\ell=15$ generated by \ref{['algorithm:SAFEM']} for the Poisson problem \ref{['eq:Hminus1']} with $L=5$ and ID (left), $L=5=K$ and GS (center), and $L=1$ (AFEM, right).
  • Figure 4: Convergence history of the error estimator $\eta_\ell(u_\ell^\kk)$ for various polynomial degrees $p$ in \ref{['algorithm:SAFEM']} applied to the Poisson problem \ref{['eq:Hminus1']}.
  • Figure 5: Final meshes $\mathcal{T}_{20}$ on level $\ell=20$ generated by \ref{['algorithm:SAFEM']} for the convection-diffusion problem \ref{['eq:convecDiff']} with $p=3$ for $K=3, L=5$ (left), $K=3, L=10$ (center), and $L=1$ (AFEM, right).
  • ...and 4 more figures

Theorems & Definitions (22)

  • Proposition 1: axioms of adaptivity
  • Proposition 2
  • Proposition 3: uniform stability of inexact Zarantonello symmetrization
  • Remark 4
  • Remark 5
  • Theorem 6: unconditional full R-linear convergence
  • Remark 7
  • Remark 8
  • Lemma 9: tail summability criterion
  • proof
  • ...and 12 more