A two-grid Adaptive Finite Element Method for the Dirichlet Boundary Control Problem Governed by Stokes Equation

2024/01/28 by Thirupathi Gudi, Gudi, Thirupathi, Ramesh Chandra Sau +1
Computer Science · Engineering · Mathematics · #65K10 #65N12 #65N15 #65N30 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Differential Equations and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.2401.15582

openalex publication_date 2024/01/28 · openalex created_date 2024/01/31 · openalex updated_date 2026/07/28

Abstract

In this article, we derive a posteriori error estimates for the Dirichlet boundary control problem governed by Stokes equation. An energy-based method has been deployed to solve the Dirichlet boundary control problem. We employ an inf-sup stable finite element discretization scheme by using P₁ elements(in the fine mesh) for the velocity and control variable and P₀ elements(in the coarse mesh) for the pressure variable. We derive an a posteriori error estimator for the state, adjoint state, and control error. The control error estimator generalizes the standard residual type estimator of the unconstrained Dirichlet boundary control problems, by additional terms at the contact boundary addressing the non-linearity. We prove the reliability and efficiency of the estimator. Theoretical results are illustrated by some numerical experiments.

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