2021/10/29 by Thirupathi Gudi, Gudi, Thirupathi, Ramesh Chandra Sau +1
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical Analysis (math.NA) #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.2110.15584
openalex publication_date 2021/10/29 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
In this paper we present a finite element analysis for a Dirichlet boundary control problem governed by the Stokes equation. The Dirichlet control is considered in a convex closed subset of the energy space H1(Ω). Most of the previous works on the Stokes Dirichlet boundary control problem deals with either tangential control or the case where the flux of the control is zero. This choice of the control is very particular and their choice of the formulation leads to the control with limited regularity. To overcome this difficulty, we introduce the Stokes problem with outflow condition and the control acts on the Dirichlet boundary only hence our control is more general and it has both the tangential and normal components. We prove well-posedness and discuss on the regularity of the control problem. The first-order optimality condition for the control leads to a Signorini problem. We develop a two-level finite element discretization by using P1 elements(on the fine mesh) for the velocity and the control variable and P0 elements (on the coarse mesh) for the pressure variable. The standard energy error analysis gives (1)/(2)+\fracδ2 order of convergence when the control is in H(3)/(2)+δ(Ω) space. Here we have improved it to (1)/(2)+δ, which is optimal. Also, when the control lies in less regular space we derived optimal order of convergence up to the regularity. The theoretical results are corroborated by a variety of numerical tests.