Error Estimates for the Numerical Approximation of Dirichlet Boundary Control for Semilinear Elliptic Equations

Eduardo Casas,Jean‐Pierre Raymond

doi:10.1137/050626600

Abstract

We study the numerical approximation of boundary optimal control problems governed by semilinear elliptic partial differential equations with pointwise constraints on the control. The control is the trace of the state on the boundary of the domain, which is assumed to be a convex, polygonal, open set in ${\mathbb R}^2$. Piecewise linear finite elements are used to approximate the control as well as the state. We prove that the error estimates are of order $O(h^{1 - 1/p})$ for some $p > 2$, which is consistent with the $W^{1 - 1/p,p}(\Gamma)$‐regularity of the optimal control.

Highlights

In this paper we study an optimal control problem governed by a semilinear elliptic equation
There are not many papers devoted to the derivation of error estimates for the discretization of control problems governed by partial differential equations; see the pioneering works by Falk [19] and Geveci [21]
In the present situation we have developed new methods, which can be used in the framework of distributed or Neumann controls to consider piecewise linear approximations

Summary

Introduction

In this paper we study an optimal control problem governed by a semilinear elliptic equation. Their procedure of proof does not work when the controls are subject to bound constraints, as considered in our problem To deal with this difficulty we assume that sufficient second order optimality conditions are satisfied. The first order conditions allow us to deduce some regularity results of the optimal control, which are necessary to derive the error estimates of the discretization. To write these conditions we have defined a discrete normal derivative for piecewise linear functions, which are solutions of some discrete equation. The sufficient conditions are proved by Casas and Mateos [9, Theorem 4.3] for distributed control problems with integral state constraints.

Ch φu

Using that