A posteriori error estimations for mixed finite-element approximations to the Navier-Stokes equations

TL;DR

This paper develops a method for estimating errors in mixed finite element solutions of the Navier-Stokes equations by reducing the problem to steady Stokes error estimation, applicable in both semi-discrete and fully discrete cases.

Contribution

It introduces a novel approach to a posteriori error estimation for Navier-Stokes equations by linking it to steady Stokes problems, including practical procedures and numerical validation.

Findings

Error estimation reduces to steady Stokes problem estimation

Practical error estimation procedure based on postprocessed approximation

Numerical experiments validate the proposed method

Abstract

A posteriori estimates for mixed finite element discretizations of the Navier-Stokes equations are derived. We show that the task of estimating the error in the evolutionary Navier-Stokes equations can be reduced to the estimation of the error in a steady Stokes problem. As a consequence, any available procedure to estimate the error in a Stokes problem can be used to estimate the error in the nonlinear evolutionary problem. A practical procedure to estimate the error based on the so-called postprocessed approximation is also considered. Both the semidiscrete (in space) and the fully discrete cases are analyzed. Some numerical experiments are provided.