Quasi-Optimality of AFEM for Distributed Optimal Control Problems of Stokes Equations: An Axiomatic Framework

TL;DR

This paper establishes the quasi-optimal convergence of an adaptive finite element method for a distributed optimal control problem governed by the Stokes equations, using an axiomatic framework and numerical validation.

Contribution

It introduces a general axiomatic framework to prove quasi-optimality of AFEM for Stokes control problems, including error estimates and convergence analysis.

Findings

Quasi-optimal convergence rates are proven for the adaptive algorithm.

Error estimates are derived under minimal regularity assumptions.

Numerical experiments confirm theoretical results on various domains.

Abstract

This paper focuses on the quasi-optimality of an adaptive nonconforming finite element method for a distributed optimal control problem governed by the Stokes equation. The nonconforming lowest order Crouzeix-Raviart element and piecewise constant spaces are used to discretise the velocity and pressure variables, respectively. The control variable is discretised using both variational and discretised approach. The error equivalence results at both continuous and discrete levels, leading to a priori and a posteriori error estimates are presented under minimal regularity assumption on optimal solutions. The quasi-optimal convergence rates of the adaptive algorithm are established based on a general axiomatic framework that includes stability, reduction, discrete reliability, and quasi-orthogonality. The theoretical findings are validated through numerical experiments on convex as well as nonconvex domains.