Mixed finite element method for a second order Dirichlet boundary control problem

TL;DR

This paper develops and analyzes mixed finite element methods for a second order Dirichlet boundary control problem, providing optimal error estimates and validating results through numerical experiments.

Contribution

It introduces a comprehensive a priori and a posteriori error analysis framework for the boundary control problem using mixed finite elements.

Findings

Optimal order a priori error estimates in energy and L2 norms.

Reliable and efficient a posteriori error estimator based on Helmholtz decomposition.

Numerical experiments confirm theoretical error bounds.

Abstract

The main aim of this article is to analyze mixed finite element method for the second order Dirichlet boundary control problem. Therein, we develop both a priori and a posteriori error analysis using the energy space based approach. We obtain optimal order a priori error estimates in the energy norm and $L^2$-norm with the help of auxiliary problems. The reliability and the efficiency of proposed a posteriori error estimator is discussed using the Helmholtz decomposition. Numerical experiments are presented to confirm the theoretical findings.