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

Devika Shylaja
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Abstract

This article focuses on the optimal control problems governed by fourth‐order linear elliptic equations with clamped boundary conditions in the framework of the Hessian discretization method (HDM). The HDM is an abstract framework that enables the convergence analysis of several numerical methods such as the conforming finite element methods, the Adini and Morley nonconforming finite element methods (ncFEMs), the 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. A companion operator for the GR method with specific property is designed. The article concludes with numerical results that illustrate the theoretical convergence rates for GR method, Adini ncFEM, and FVM.
利用 Hessian 离散法对受四阶线性椭圆方程支配的优化控制问题进行数值分析
这篇文章在黑森离散化方法(HDM)的框架内,重点研究了具有箝位边界条件的四阶线性椭圆方程所控制的最优控制问题。HDM 是一个抽象框架,可以对多种数值方法进行收敛分析,如符合有限元方法、Adini 和 Morley 非符合有限元方法(ncFEMs)、基于梯度恢复(GR)算子的方法以及有限体积方法(FVMs)。为状态变量、邻接变量和控制变量建立了基本误差估计和超收敛结果。文章还设计了一个具有特定性质的 GR 方法辅助算子。文章最后以数值结果说明了 GR 方法、Adini ncFEM 和 FVM 的理论收敛率。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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