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

TL;DR

This paper introduces a novel error analysis framework for finite element methods applied to elliptic boundary control problems with Neumann boundary conditions and pointwise constraints, accommodating both smooth and rough coefficients.

Contribution

It provides a new error analysis applicable to standard and multiscale finite element methods for elliptic control problems with pointwise constraints.

Findings

Applicable to smooth coefficient cases

Extends to multiscale methods with rough coefficients

Enhances understanding of finite element error behavior

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.