# Adaptive Uzawa algorithm for the Stokes equation

## TL;DR

This paper introduces an adaptive finite element method based on the Uzawa algorithm for the Stokes system, achieving optimal convergence rates without relying on data discretization or interior node properties.

## Contribution

It presents a novel adaptive Uzawa algorithm that ensures linear convergence with optimal algebraic rates, avoiding the need for data discretization and interior node assumptions.

## Key findings

- Proves linear convergence of the adaptive method.
- Achieves optimal algebraic convergence rates.
- Does not require discretization of data or interior node property.

## Abstract

Based on the Uzawa algorithm, we consider an adaptive finite element method for the Stokes system. We prove linear convergence with optimal algebraic rates for the residual estimator (which is equivalent to the total error), if the arising linear systems are solved iteratively, e.g., by PCG. Our analysis avoids the use of discrete efficiency of the estimator. Unlike prior work, our adaptive Uzawa algorithm can thus avoid to discretize the given data and does not rely on an interior node property for the refinement.