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

TL;DR

This paper provides a rigorous convergence analysis of the smoothed adaptive finite element method (S-AFEM), demonstrating its unconditional linear convergence and optimal complexity, while reducing computational costs compared to classical AFEM.

Contribution

The paper proves the first rigorous convergence and complexity results for S-AFEM, showing it retains AFEM guarantees with lower computational effort.

Findings

S-AFEM achieves unconditional full R-linear convergence.

S-AFEM attains optimal convergence rates with respect to computational cost.

Numerical experiments confirm theoretical predictions and efficiency gains.

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.