A Posteriori Estimates of Taylor-Hood Element for Stokes Problem Using Auxiliary Subspace Techniques

TL;DR

This paper introduces a hierarchical basis a posteriori error estimator for the Stokes problem that significantly reduces computational costs and is validated through numerical simulations demonstrating its effectiveness.

Contribution

It proposes a novel a posteriori error estimator based on auxiliary subspace techniques for the Taylor-Hood element in Stokes problems, with proven reliability and efficiency.

Findings

Reduces computational cost by solving only two global diagonal systems.

Provides reliable upper and lower bounds for the error estimator.

Numerical results confirm robustness and effectiveness.

Abstract

Based on the auxiliary subspace techniques, a hierarchical basis a posteriori error estimator is proposed for the Stokes problem in two and three dimensions. For the error estimator, we need to solve only two global diagonal linear systems corresponding to the degree of freedom of velocity and pressure respectively, which reduces the computational cost sharply. The upper and lower bounds up to an oscillation term of the error estimator are also shown to address the reliability of the adaptive method without saturation assumption. Numerical simulations are performed to demonstrate the effectiveness and robustness of our algorithm.