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A New Error Analysis for Finite Element Methods for Elliptic Neumann Boundary Control Problems with Pointwise Control Constraints

Susanne C. Brenner, Li-yeng Sung

Abstract

We present a new error analysis for finite element methods for a linear-quadratic elliptic optimal control problem with Neumann boundary control and pointwise control constraints. It can be applied to standard finite element methods when the coefficient s in the elliptic operator are smooth and also to multiscale finite element methods when the coefficients are rough.

A New Error Analysis for Finite Element Methods for Elliptic Neumann Boundary Control Problems with Pointwise Control Constraints

Abstract

We present a new error analysis for finite element methods for a linear-quadratic elliptic optimal control problem with Neumann boundary control and pointwise control constraints. It can be applied to standard finite element methods when the coefficient s in the elliptic operator are smooth and also to multiscale finite element methods when the coefficients are rough.

Paper Structure

This paper contains 13 sections, 5 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. The Continuous Problem
  3. The Discrete Problem
  4. An Abstract Error Analysis
  5. Application to Problems with Smooth Coefficients
  6. Regularity Results for the Continuous Problem
  7. Results for the $P_1$ Finite Element Method
  8. Variational Discretization Concept
  9. Application to Problems with Rough Coefficients
  10. A Standard $P_1$ Finite Element Method
  11. Multiscale Finite Element Methods
  12. Concluding Remarks
  13. Some Estimates for $\bar{y}$, $\bar{u}$ and $\bar{p}$

Key Result

Lemma 3.3

Let $V$ be a closed subspace of ${H^1(\Omega)}$, $r\in{L_2(\Omega)}$ and $g\in{L_2(\Gamma)}$. If $v\in V$ is defined by then we have where $C_{\rm Tr}$ is the positive constant that appears in the trace inequality

Theorems & Definitions (16)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 3.1
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Theorem 4.1
  • proof
  • Lemma 6.1
  • ...and 6 more