Residual-Based a posteriori error estimation for hp-adaptive finite element methods for the Stokes equations

TL;DR

This paper develops a residual-based a posteriori error estimator for hp-adaptive finite element methods applied to the steady state Stokes equations, enhancing adaptive fluid flow simulations.

Contribution

It extends classical a posteriori error estimation techniques to hp-adaptive methods for the Stokes problem, establishing reliability and efficiency.

Findings

The error estimator is reliable and efficient.

Numerical experiments demonstrate improved adaptive hp-FEM performance.

The method is applicable to incompressible fluid flow simulations.

Abstract

We derive a residual-based a posteriori error estimator for the conforming hp-Adaptive Finite Element Method (hp-AFEM) for the steady state Stokes problem describing the slow motion of an incompressible fluid. This error estimator is obtained by extending the idea of a posteriori error estimation for the classical $h$-version of AFEM. We also establish the reliability and efficiency of the error estimator. The proofs are based on the well-known Clement-type interpolation operator introduced in 2005 in the context of the hp-AFEM. Numerical experiments show the performance of an adaptive hp-FEM algorithm using the proposed a posteriori error estimator.