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.

💡 Research Summary

This paper provides the first rigorous convergence analysis of the Smoothed Adaptive Finite Element Method (S‑AFEM) originally introduced by Mulita, Giani, and Heltai (SIAM J. Sci. Comput., 2021). Classical Adaptive Finite Element Methods (AFEM) solve the discrete Galerkin problem exactly on every refinement level, compute a residual‑based error estimator, mark elements, and refine the mesh. The exact solve step is usually the most expensive part of the loop. S‑AFEM reduces this cost by performing an accurate solve only on every L‑th mesh level, while on the intermediate levels it applies a fixed, small number J of inexpensive smoothing iterations. The authors prove that, despite this inexactness, S‑AFEM inherits all the abstract convergence guarantees of AFEM and even improves computational efficiency.

The analysis starts with a general second‑order linear elliptic PDE (possibly non‑symmetric) posed on a bounded polyhedral domain. The weak formulation satisfies the Lax–Milgram conditions, and the standard conforming Lagrange finite element spaces of degree p are used. A residual‑based error estimator η is employed, and the four axioms of adaptivity (stability, reduction, reliability, quasi‑monotonicity) are recalled. In addition, a quasi‑orthogonality property (A4) is used to handle the lack of a true Pythagorean identity for non‑symmetric problems.

The core of the theoretical contribution lies in the abstract modeling of the smoother. A smoother is represented by an iteration operator Ψ_H : X_H → X_H. Two mesh‑independent properties are required:

1. Uniform Stability (US): ‖u*_H – Ψ_H(v_H)‖ ≤ C_alg ‖u*_H – v_H‖ for all v_H ∈ X_H.
2. Uniform Contraction (UC): the same inequality with C_alg = q_alg < 1.

These conditions are extremely mild; they hold for Richardson, damped Jacobi, Gauss–Seidel, Conjugate Gradient, preconditioned CG, multigrid, and many other standard solvers. For non‑symmetric problems the authors employ a Zarantonello symmetrization step Z_δ^H, which maps a non‑symmetric residual to a symmetric one. By applying a uniformly stable smoother to the symmetric system and iterating it J times, they obtain a uniformly stable (and, for sufficiently large J, uniformly contractive) smoother for the original non‑symmetric problem. The constants involved depend only on the ellipticity and continuity constants of the PDE and on the chosen δ.

Algorithm A (the formal S‑AFEM algorithm) proceeds as follows: on a mesh T_ℓ, if ℓ is a “solve level” (i.e., ℓ mod L = 0), the exact Galerkin solution u*_ℓ is computed; otherwise, the current approximation is updated by applying Ψ_ℓ J times (the SMOOTH module). The ESTIMATE step evaluates η_ℓ using either the exact or the smoothed solution. The MARK step uses Dörfler marking, but on intermediate levels a cardinality‑control mechanism limits the number of marked elements, ensuring that the refinement cost remains linear in the number of elements. REFINE then performs newest‑vertex bisection (NVB) to obtain a conforming refined mesh.

The convergence analysis is split into two main results. Theorem 6 establishes unconditional R‑linear convergence of the quasi‑error \