Residual-based a posteriori error estimates for hp finite element solutions of semilinear Neumann boundary optimal control problems

Zuliang Lu,Hongyan Liu,Shuhua Zhang,Chuanjuan Hou

doi:10.1186/s13661-016-0562-2

Abstract

In this paper, we investigate residual-based a posteriori error estimates for the hp finite element approximation of semilinear Neumann boundary elliptic optimal control problems. By using the hp finite element approximation for both the state and the co-state and the hp discontinuous Galerkin finite element approximation for the control, we derive a posteriori error bounds in $L^{2}$ - $H^{1}$ norms for the Neumann boundary optimal control problems governed by semilinear elliptic equations. We also give $L^{2}$ - $L^{2}$ a posteriori error estimates for the optimal control problems. Such estimates, which are apparently not available in the literature, can be used to construct reliable adaptive finite element approximations for the semilinear Neumann boundary optimal control problems.

Highlights

1 Introduction In this paper, we study residual-based a posteriori error estimates for the hp finite element approximation of semilinear Neumann boundary optimal control problems
In [ ], for the constrained optimal control problem governed by linear elliptic equations, the authors have derived a posteriori error estimates for the hp finite element solutions
2 Finite element methods of boundary optimal control we study the hp finite element approximation of semilinear convex optimal control problems where the control appears in the Neumann boundary conditions

Summary

Introduction

1 Introduction In this paper, we study residual-based a posteriori error estimates for the hp finite element approximation of semilinear Neumann boundary optimal control problems. In [ ], the authors studied a posteriori error estimates for adaptive finite element discretizations of boundary control problems. In [ ], for the constrained optimal control problem governed by linear elliptic equations, the authors have derived a posteriori error estimates for the hp finite element solutions. Inspired by the work of [ ], we consider a posteriori error estimates in L -H norms and L -L norms for hp finite element solutions of general semilinear Neumann boundary optimal control problems.

Results

Conclusion