An error analysis of discontinuous finite element methods for the optimal control problems governed by Stokes equation

TL;DR

This paper develops an abstract error analysis framework for discontinuous finite element methods applied to optimal control problems governed by the stationary Stokes equation, providing both a priori and a posteriori error estimates.

Contribution

It introduces a novel error analysis framework for discontinuous finite element methods in Stokes control problems, including minimal regularity cases and practical error estimators.

Findings

Derives optimal order a priori error estimates for velocity and pressure.

Establishes reliable and efficient a posteriori error estimators.

Numerical experiments confirm the theoretical results.

Abstract

In this paper, an abstract framework for the error analysis of discontinuous finite element method is developed for the distributed and Neumann boundary control problems governed by the stationary Stokes equation with control constraints. {\it A~priori} error estimates of optimal order are derived for velocity and pressure in the energy norm and the $L^2$-norm, respectively. Moreover, a reliable and efficient {\it a~posteriori} error estimator is derived. The results are applicable to a variety of problems just under the minimal regularity possessed by the well-posedness of the problem. In particular, we consider the abstract results with suitable stable pairs of velocity and pressure spaces like as the lowest-order Crouzeix-Raviart finite element and piecewise constant spaces, piecewise linear and constant finite element spaces. The theoretical results are illustrated by the numerical experiments.