Crouzeix-Raviart Finite Element Approximation of Dirichlet Boundary Control Problems with Piecewise Constant Controls

TL;DR

This paper studies boundary control problems for the Poisson equation using Crouzeix-Raviart finite elements and piecewise constant controls, providing error estimates for the discretization.

Contribution

It introduces a novel discretization approach combining Crouzeix-Raviart elements with piecewise constant controls and derives optimal error estimates.

Findings

Established an optimal order a priori error estimate for the control variable.

Demonstrated the effectiveness of the ultra-weak formulation with Crouzeix-Raviart elements.

Analyzed boundary control problems on convex polygonal domains.

Abstract

This article examines the Dirichlet boundary control problem governed by the Poisson equation, where the control variables are square integrable functions defined on the boundary of a two-dimensional bounded, convex, polygonal domain. It employs an ultra-weak formulation and utilizes Crouzeix-Raviart finite elements to discretize the state variable, while employing piecewise constants for the control variable discretization. Furthermore, it establishes an optimal order a priori error estimate for the control variable.