Quasi-Optimality of AFEM for Distributed Optimal Control Problems of Stokes Equations: An Axiomatic Framework
This work addresses the efficient numerical solution of optimal control problems in fluid dynamics, representing an incremental improvement in adaptive methods for Stokes equations.
The paper tackled the quasi-optimality of an adaptive finite element method for distributed optimal control problems governed by the Stokes equation, establishing quasi-optimal convergence rates through an axiomatic framework and validating them with numerical experiments on convex and nonconvex domains.
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.