双调和方程控制的含狄拉克测度最优控制问题的自适应有限元方法的拟最优性
Quasi-optimality of adaptive FEM for optimal control problems involving Dirac measures governed by biharmonic equation
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中文总结 AI 辅助
该研究针对双调和方程控制的含狄拉克测度最优控制问题，采用非协调Morley有限元结合伴随算子修改右端项的方法，确立了自适应有限元方法的拟最优性与最优收敛速率，相关数值实验验证了理论结果。
中文摘要 AI 辅助
本文确立了双调和方程控制的一类含狄拉克测度最优控制问题的自适应非协调有限元方法的拟最优性。采用非协调Morley有限元对状态变量和伴随变量进行离散；通过将Morley有限元映射到协调空间的伴随算子修改右端项，以克服处理右端项点源的挑战。推导了该最优控制问题的先验和后验误差估计，并利用公理框架建立了自适应有限元方法的最优收敛速率，具体通过证明稳定性、缩减性、离散可靠性和拟正交性等关键性质实现。对三类最优控制问题的数值实验进行了广泛讨论，验证了理论结果。
英文摘要
This article establishes the quasi-optimality of adaptive nonconforming finite element methods for a class of optimal control problems involving Dirac measures governed by the biharmonic equation. The nonconforming Morley finite elements are employed for discretising both the state and adjoint variables. A modified right-hand side through a companion operator that maps Morley finite elements toaconformingspacehelpstoovercomethechallengeinhandlingpointsourcesontheright-hand side. A priori and a posteriori error estimates for the optimal control problems are derived. Further,optimal convergence rates for adaptive finite element methods are established using an axiomatic framework: by proving key properties such as stability, reduction, discrete reliability, and quasi-orthogonality. Numerical experiments for three types of optimal control problems are discussed extensively and they validate the theoretical results.
