# Quasi-optimal mesh sequence construction through Smoothed Adaptive Finite Element Method

## TL;DR

This paper introduces S-AFEM, a smoothing-based adaptive finite element method that achieves similar refinement patterns to classical AFEM but with significantly reduced computational cost, validated through analysis and experiments.

## Contribution

The paper presents a novel S-AFEM algorithm that replaces algebraic solutions with smoothing iterations, offering efficiency gains while maintaining accuracy.

## Key findings

- Refinement patterns are similar to classical AFEM.
- S-AFEM reduces computational cost significantly.
- Numerical experiments confirm efficiency and speedup.

## Abstract

We propose a new algorithm for Adaptive Finite Element Methods (AFEMs) based on smoothing iterations (S-AFEM), for linear, second-order, elliptic partial differential equations (PDEs). The algorithm is inspired by the ascending phase of the V-cycle multigrid method: we replace accurate algebraic solutions in intermediate cycles of the classical AFEM with the application of a prolongation step, followed by the application of a smoother. Even though these intermediate solutions are far from the exact algebraic solutions, their a-posteriori error estimation produces a refinement pattern that is substantially equivalent to the one that would be generated by classical AFEM, at a considerable fraction of the computational cost. We provide a qualitative analysis of how the error propagates throughout the algorithm, and we present a series of numerical experiments that highlight the efficiency and the computational speedup of S-AFEM.