Numerical analysis of optimal control problems governed by fourth-order linear elliptic equations using the Hessian discretisation method

TL;DR

This paper analyzes optimal control problems governed by fourth-order linear elliptic equations using the Hessian discretisation method, providing convergence analysis, error estimates, and numerical validation for various numerical schemes.

Contribution

It introduces a unified framework (HDM) for analyzing multiple numerical schemes for fourth-order elliptic control problems, including new error and superconvergence results.

Findings

Established basic error estimates for state, adjoint, and control variables.

Proved superconvergence results within the HDM framework.

Numerical experiments confirmed theoretical convergence rates.

Abstract

This paper focusses on the optimal control problems governed by fourth-order linear elliptic equations with clamped boundary conditions in the framework of the Hessian discretisation method (HDM). The HDM is an abstract framework that enables the convergence analysis of numerical methods through a quadruplet known as a Hessian discretisation (HD) and three core properties of HD. The HDM covers several numerical schemes such as the conforming finite element methods, the Adini and Morley non-conforming finite element methods (ncFEMs), method based on gradient recovery (GR) operators and the finite volume methods (FVMs). Basic error estimates and superconvergence results are established for the state, adjoint and control variables in the HDM framework. The article concludes with numerical results that illustrates the theoretical convergence rates for the GR method, Adini ncFEM and FVM.