Error estimates of $hp$-finite element method for elliptic optimal control problems with robin boundary

TL;DR

This paper develops both a priori and a posteriori error estimates for the $hp$-finite element method applied to elliptic optimal control problems with Robin boundary conditions, validated through numerical examples.

Contribution

It introduces new error estimation techniques for $hp$ finite element methods in elliptic control problems with Robin boundaries, combining Clément-type and Scott-Zhang-type approaches.

Findings

Error estimates are validated by numerical examples.

The proposed estimators accurately predict errors in elliptic control problems.

The methods improve understanding of $hp$-FEM performance with Robin boundary conditions.

Abstract

A priori and a posteriori error analysis of $hp$ finite element method for elliptic control problem with Robin boundary condition and boundary observation are presented. are presented. Through the Cl\'ement-type approach and the construction of an auxiliary system, we derived a priori error estimates for the elliptic optimal control problem. Residual-based a posteriori error estimates are derived based on the well-known Scott-Zhang-type quasi-interpolation and coupled state-control approximations, thus establishing an a posteriori error estimator for the $hp$ finite element method. The numerical example demonstrates the accuracy of error estimation for the elliptic optimal control problems with Robin boundary.