Error estimates for the postprocessing approach applied to Neumann boundary control problems in polyhedral domains
Error estimates for the postprocessing approach applied to Neumann boundary control problems in polyhedral domains
复制标题
应用于多面体域中诺依曼边界控制问题的后处理方法的误差估计
DOI：
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发表时间：
2018
期刊：
影响因子：
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通讯作者：
J. Pfefferer
中科院分区：
文献类型：
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作者：
T. Apel;M. Winkler;J. Pfefferer
本文讨论多面体区域上Neumann边界控制问题的有限元逼近的误差估计。特别强调的是奇异性包含在解决方案中的计算域的边缘和角落。因此，我们在加权Sobolev空间中使用定制的正则性结果，从而可以导出局部细化网格的尖锐收敛结果。第一个主要结果是一个最佳的误差估计线性有限元逼近的边界上的L（）-范数的准均匀和各向同性细化网格。随后，近似的Neumann控制问题使用后处理方法进行了研究，即，首先计算一个完全离散的解决方案，分段线性状态和co-state，分段常数控制，然后，通过逐点评估的离散最优性条件得到一个改进的控制。结果表明，二次收敛对数因子实现了这种控制近似，如果奇点是足够弱或网格序列进行适当的细化。
This paper deals with error estimates for the finite element approximation of Neumann boundary control problems in polyhedral domains. Special emphasis is put on singularities contained in the solution as the computational domain has edges and corners. Thus, we use tailored regularity results in weighted Sobolev spaces which allow to derive sharp convergence results for locally refined meshes. The first main result is an optimal error estimate for linear finite element approximations on the boundary in the L( )-norm for both quasi-uniform and isotropically refined meshes. Later, the approximations of Neumann control problems using the postprocessing approach are investigated, that is, first a fully discrete solution with piecewise linear state and co-state, and piecewise constant controls, is computed and afterwards, an improved control by a pointwise evaluation of the discrete optimality condition is obtained. It is shown that quadratic convergence up to logarithmic factors is achieved for this control approximation if either the singularities are weak enough or the sequence of meshes is refined appropriately.