AI 中文总结
研究泊松问题有限元离散化的r细化，提出标准优化算法，将形状优化技术用于标准残差误差估计器，通过数值实验验证了该方法在自适应r细化形状优化中的有效性。
AI 中文摘要
我们考虑泊松问题有限元离散化的r细化。r细化的目标是重新定位计算网格的节点，以更好地逼近有限元误差。由于网格在移动，它自然与形状优化方法相关联。我们为这个r细化过程提出了一种标准优化算法，并表明如果试图最小化实际误差（通常无法计算），有一个会终止的算法。这项工作最新颖之处在于将形状优化技术应用于标准残差误差估计器，对于一维泊松方程，该估计器是关于网格可微的泛函。为说明该方法，给出了一些数值实验，验证了方法的有效性。
英文摘要
We consider $r$-refinement for the finite element discretisation of a Poisson problem. The goal of $r$-refinement is to reposition the nodes of a computational mesh in order to better approximate the finite element error. Since the mesh is being moved, it naturally becomes linked to shape optimisation methods. We propose a standard optimisation algorithm for this $r$-refinement procedure and show that if one seeks to minimise the actual error - which one cannot generally calculate - one has an algorithm which will terminate. The most novel aspect of this work is to apply shape optimisation techniques to the a standard residual error estimator, which is a functional that is differentiable with respect to the mesh, when considered for the Poisson equation in one dimension. To illustrate the approach, a number of numerical experiments are presented, which verify the efficacy of the method.
