Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Bilinear optimal control for the Stokes-Brinkman equations: a priori and a posteriori error analyses

Alejandro Allendes, Gilberto Campaña, Enrique Otarola

Abstract

We analyze a bilinear optimal control problem for the Stokes--Brinkman equations: the control variable enters the state equations as a coefficient. In two- and three-dimensional Lipschitz domains, we perform a complete continuous analysis that includes the existence of solutions and first- and second-order optimality conditions. We also develop two finite element methods that differ fundamentally in whether the admissible control set is discretized or not. For each of the proposed methods, we perform a convergence analysis and derive a priori error estimates; the latter under the assumption that the domain is convex. Finally, assuming that the domain is Lipschitz, we develop an a posteriori error estimator for each discretization scheme, obtain a global reliability bound, and investigate local efficiency estimates.

Bilinear optimal control for the Stokes-Brinkman equations: a priori and a posteriori error analyses

Abstract

We analyze a bilinear optimal control problem for the Stokes--Brinkman equations: the control variable enters the state equations as a coefficient. In two- and three-dimensional Lipschitz domains, we perform a complete continuous analysis that includes the existence of solutions and first- and second-order optimality conditions. We also develop two finite element methods that differ fundamentally in whether the admissible control set is discretized or not. For each of the proposed methods, we perform a convergence analysis and derive a priori error estimates; the latter under the assumption that the domain is convex. Finally, assuming that the domain is Lipschitz, we develop an a posteriori error estimator for each discretization scheme, obtain a global reliability bound, and investigate local efficiency estimates.
Paper Structure (37 sections, 35 theorems, 209 equations, 9 figures, 3 algorithms)

This paper contains 37 sections, 35 theorems, 209 equations, 9 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Notation and preliminary remarks
  3. Notation
  4. The Stokes--Brinkman problem
  5. Regularity results
  6. Differentiability properties of the map $\mathfrak{u} \to (\boldsymbol{\mathsf{y}},\mathfrak{p})$
  7. The optimal control problem
  8. Weak formulation
  9. Existence of solutions
  10. First-order optimality conditions
  11. Second-order optimality conditions
  12. Necessary second-order optimality conditions
  13. Sufficient second-order optimality conditions
  14. Finite element approximation of the state equations
  15. Finite element spaces
  16. ...and 22 more sections

Key Result

Theorem 1

\newlabeltheorem_regularity_Linfty0 Let $\boldsymbol{\mathfrak{f}}$ and $\mathfrak{u}$ be in $L^2(\Omega)$. If the domain $\Omega$ is Lipschitz and $(\boldsymbol{\mathsf{y}},\mathfrak{p})$ is a solution of problem eq:brinkman_problem, then there exists $\kappa$ such that $(\boldsymbol{\mathsf{y}},\ where the hidden constants depend on $\| u \|_{L^{2}(\Omega)}$. In particular, $\boldsymbol{\mathsf{

Figures (9)

  • Figure 1: Initial meshes for $\Omega=(0,1)^{2}$ (left) and $\Omega=(-1, 1) \times (-1, 1) \setminus [0, 1] \times [-1,0]$ (right).
  • Figure 2: Example 1. Experimental convergence rates for the fully discrete scheme with uniform and adaptive refinement: state and adjoint state velocity errors (A.1); state and adjoint state pressure errors (A.2); control errors (A.3); total errors (A.4); state error estimators (A.5); adjoint state error estimators (A.6); control error estimators (A.7); total error estimators (A.8); and effectivity indices (A.9).
  • Figure 3: Example 1. Experimental convergence rates for the semidiscrete scheme with uniform and adaptive refinement: state and adjoint state velocity errors (B.1); state and adjoint state pressure errors (B.2); control errors (B.3); total errors (B.4); state error estimators (B.5); adjoint state error estimators (B.6); total error estimators (B.7); and effectivity indices (B.8).
  • Figure 4: Example 1. For the fully discrete scheme we present the mesh obtained after 35 iterations of adaptive refinement (18.504 triangles and 9.285 vertices), the discrete boundary of the active sets, where the blue line corresponds to the one for $\mathsf{a}=0.1$, and the red line corresponds to the one for $\mathsf{b}=0.2$ (C.1); the exact boundary of the corresponding active sets (C.2), the discrete optimal adjoint velocity $|\bar{\mathbf{z}}_{h}|$ (C.3); and the discrete optimal control $\bar{u}_{h}$ (C.4).
  • Figure 5: Example 1. For the semidiscrete scheme we present the mesh obtained after 35 iterations of adaptive refinement (18.504 triangles and 9.285 vertices), the discrete boundary of the active sets, where the blue line corresponds to the one for $\mathsf{a}=0.1$, and the red line corresponds to the one for $\mathsf{b}=0.2$ (D.1); the exact boundary of the corresponding active sets (D.2); the discrete optimal adjoint velocity $|\bar{\mathbf{z}}_{h}|$ (D.3); and the discrete optimal control $\bar{\mathsf{u}}=\Pi_{[\mathsf{a},\mathsf{b}]}(\alpha^{-1}\bar{\mathbf{y}}_{h}\cdot\bar{\mathbf{z}}_{h})$ (D.4).
  • ...and 4 more figures

Theorems & Definitions (71)

  • Theorem 1: Sobolev and Hölder regularity
  • Proof 1
  • Theorem 2: $\mathbf{H}^2(\Omega)\times H^1(\Omega)$-regularity
  • Proof 2
  • Lemma 3: embedding
  • Proof 3
  • Theorem 4: differentiability properties of $\mathfrak{u} \to (\boldsymbol{\mathsf{y}},\mathfrak{p})$
  • Proof 4
  • Remark 5: the admissible control set
  • Remark 6: the state equations
  • ...and 61 more