Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

New Error Estimates for An Elliptic Distributed Optimal Control Problem with Pointwise Control Constraints

Susanne C. Brenner, Li-yeng Sung

Abstract

We derive error estimates for a linear-quadratic elliptic distributed optimal control problem with pointwise control constraints that can be applied to standard finite element methods and multiscale finite element methods.

New Error Estimates for An Elliptic Distributed Optimal Control Problem with Pointwise Control Constraints

Abstract

We derive error estimates for a linear-quadratic elliptic distributed optimal control problem with pointwise control constraints that can be applied to standard finite element methods and multiscale finite element methods.
Paper Structure (13 sections, 4 theorems, 88 equations)

This paper contains 13 sections, 4 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. The Continuous Problem
  3. Bounds for $\|\bar{y}-y_d\|_{L_2(\Omega)}$ and $\|\bar{u}\|_{L_2(\Omega)}$
  4. Bounds for $|\bar{u}|_{H^1(\Omega)}$ and $|\bar{p}|_{H^1(\Omega)}$
  5. The Lagrange Multiplier $\lambda$
  6. The Approximation Problem
  7. Error Estimates
  8. Estimate for the $L_2$ Errors
  9. Estimate for the Energy Errors
  10. Applications
  11. Standard Finite Element Methods
  12. Multiscale Finite Element Methods
  13. Conclusions

Key Result

Lemma 3.1

Let $g\in{L_2(\Omega)}$ and $v_*\in V_*$ satisfy We have

Theorems & Definitions (13)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 2.1
  • Lemma 3.1
  • proof
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • ...and 3 more