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首页> 外文期刊>Journal of Applied Mathematics and Computing >Two fast finite difference schemes for elliptic Dirichlet boundary control problems
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Two fast finite difference schemes for elliptic Dirichlet boundary control problems

机译：两个快速有限差分方案，用于椭圆Dirichlet边界控制问题

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摘要

In this paper, we propose and analyze two standard finite difference schemes (called Scheme I and Scheme II) for discretizing the first-order necessary optimality systems, which characterize the optimal solutions of Dirichlet boundary control problems governed by elliptic equations.We proved that the proposed schemes are uniformly stable on a uniform mesh, which implies a second-order and first-order convergence of the Scheme I and Scheme II, respectively, provided the optimal solutions have the required regularity. The resulting symmetric indefinite sparse linear systems are solved by the preconditioned GMRES iterative solver with a fast (FFT-based) constraint preconditioner, which numerically shows a mesh-independent convergence rate. Numerical examples, including the case with less regular solutions, are presented to validate our theoretical analysis and demonstrate the promising approximation accuracy and computational efficiency of our proposed schemes and preconditioned iterative solver, respectively. Our developed fast finite difference schemes achieve a comparable order of convergence as the other available schemes in the literature.
机译：在本文中，我们提出并分析了两个标准有限差分方案（称为方案I和方案II），用于离散化一阶必要的最优性系统，该系统表征了椭圆方程所控制的Dirichlet边界控制问题的最佳解决方案。我们证明了提出的方案在均匀的网格上均匀稳定，其暗示了方案I和方案II的二阶和一阶收敛，只要最佳解决方案具有所需的规则性。由此产生的对称的无限稀疏线性系统由预处理的GMRES迭代求解器解决了快速（基于FFT的）约束预处理器，其数值上显示了网状无关的收敛速率。介绍了数值示例，包括具有较少常规解决方案的情况，以验证我们的理论分析，并分别展示了我们提出的方案和预处理迭代求解器的有希望的近似准确度和计算效率。我们开发的快速有限差分方案可实现可比的收敛顺序作为文献中的其他可用方案。

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