Christoph Lietz | Searcharxiv

arXiv2026

Quasi-optimal complexity of iterative Galerkin methods driven by an elliptic reconstruction error estimator

We study an iterative Galerkin method for quasilinear elliptic problems in the Browder-Minty setting. The resulting discrete nonlinear systems are solved by linearization via a (damped) Zarantonello iteration. Unlike prior work, adaptive mesh refinement is driven by an elliptic reconstruction error estimator, which is natural in the sense that the a posteriori bounds for the linearization and discretization errors are well separated. For this setting, we present the first comprehensive convergence analysis of the corresponding algorithm. We prove unconditional full R-linear convergence of a suitable quasi-error that combines linearization and discretization errors. For sufficiently small adaptivity parameters, we further establish optimal convergence rates with respect to the number of degrees of freedom and quasi-optimal complexity, i.e., optimal convergence rates with respect to the overall computational cost. Numerical experiments underpin the theoretical findings.

math.NA↗

arXiv2026

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

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.

math.NA↗