Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Thirupathi Gudi, Ramesh Chandra Sau

TL;DR

The paper addresses the Dirichlet boundary control problem for the Stokes equations with Signorini-type constraints. It develops an energy-space formulation and a two-grid inf-sup stable finite element discretization, using velocity and control in $P_1$ on the fine grid and pressure in $P_0$ on the coarse grid. A novel a posteriori error estimator is derived for the state, adjoint state, and control, based on a Galerkin functional and incorporating complementarity terms to handle the nonlinear boundary contact, with proven reliability and efficiency. Numerical experiments on smooth and nonconvex domains demonstrate the estimator's effectiveness and yield optimal adaptive convergence rates.

Abstract

In this article, we derive \textit{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 $\mathbf{P}_1$ elements(in the fine mesh) for the velocity and control variable and $P_0$ elements(in the coarse mesh) for the pressure variable. We derive an \textit{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.

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

TL;DR

The paper addresses the Dirichlet boundary control problem for the Stokes equations with Signorini-type constraints. It develops an energy-space formulation and a two-grid inf-sup stable finite element discretization, using velocity and control in on the fine grid and pressure in on the coarse grid. A novel a posteriori error estimator is derived for the state, adjoint state, and control, based on a Galerkin functional and incorporating complementarity terms to handle the nonlinear boundary contact, with proven reliability and efficiency. Numerical experiments on smooth and nonconvex domains demonstrate the estimator's effectiveness and yield optimal adaptive convergence rates.

Abstract

In this article, we derive \textit{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 elements(in the fine mesh) for the velocity and control variable and elements(in the coarse mesh) for the pressure variable. We derive an \textit{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.
Paper Structure (8 sections, 9 theorems, 111 equations, 5 figures)

This paper contains 8 sections, 9 theorems, 111 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Continuous Problem
  3. Notation
  4. Dirichlet Control Problem
  5. Discrete Problem
  6. A posteriori Error Analysis
  7. Numerical Experiments
  8. Conclusions

Key Result

Proposition 2.1

There exists a unique solution $(\mathbf{u},p,\mathbf{y})\in \mathbf{H}_{D}^1(\Omega)\times L^2(\Omega)\times \mathbf{Q}_{ad}$ for the Dirichlet control problem min:j and there exists an adjoint state $(\bm{\phi},r)\in \mathbf{V}\times L^2(\Omega)$ satisfying where $a(\mathbf{w},\mathbf{z})= \int_{\Omega} \nabla{\mathbf{w}}:\nabla{\mathbf{z}} {\rm~dx}$ , $b(\mathbf{z},p)= - \int_{\Omega} p\nabla

Figures (5)

  • Figure 3.1: Here $T_-$ and $T_{+}$ are the two neighboring triangles that share the edge $e=\partial T_-\cap\partial T_+$ with initial node $A$ and end node $B$ and unit normal $\mathbf{n}_e$. The orientation of $\mathbf{n}_e = \mathbf{n}_{-} = -\mathbf{n}_{+}$ equals the outer normal of $T_{-}$, and hence, points into $T_{+}$.
  • Figure 4.1: Subgrid of boundary patch $\gamma_{p,C}$
  • Figure 5.1:
  • Figure 5.2: Convergence history on uniform mesh (L-shape domain).
  • Figure 5.3:

Theorems & Definitions (18)

  • Proposition 2.1
  • Remark 2.2
  • Proposition 3.1: Discrete Optimality System
  • Theorem 4.1: Energy error estimate of control and $L^2$-estimate of velocity
  • proof
  • Theorem 4.2: Energy error estimate of velocity
  • proof
  • Theorem 4.3: Energy error estimate of adjoint velocity
  • proof
  • Theorem 4.4: Error estimate of pressure and adjoint pressure
  • ...and 8 more