论文标题
在同几何分析中观察有限的对流扩散反应方程的最佳控制对流 - 扩散反应方程的最佳控制和误差估计值
Robust preconditioning and error estimates for optimal control of the convection-diffusion-reaction equation with limited observation in Isogeometric analysis
论文作者
论文摘要
在本文中，我们分析了一个优化问题，其中有限的观察值受对流 - 扩散 - 反应方程。通过Schur补体方法的启发，我们得出了连续的规范，可以分析适当的误差分析和随后的误差分析和对问题参数相对于鲁棒的预处理。我们为INF-SUP稳定离散化提供条件，并对具有恒定对流的框域呈现一个这样的离散化。我们还为此离散化提供了先验错误估计。预处理需要解决第四阶问题。因此，我们将等几何分析用作离散化方法。为了有效地实现预处理，我们考虑使用标准的高斯 - 西德尔（Gauss-seidel）和新的宏观高斯 - 西德尔（Gauss-Seidel）更加顺滑的几何多机。后者更顺畅地相对于几何图和多项式度提供了良好的结果。
In this paper we analyze an optimization problem with limited observation governed by a convection--diffusion--reaction equation. Motivated by a Schur complement approach, we arrive at continuous norms that enable analysis of well-posedness and subsequent derivation of error analysis and a preconditioner that is robust with respect to the parameters of the problem. We provide conditions for inf-sup stable discretizations and present one such discretization for box domains with constant convection. We also provide a priori error estimates for this discretization. The preconditioner requires a fourth order problem to be solved. For this reason, we use Isogeometric Analysis as a method of discretization. To efficiently realize the preconditioner, we consider geometric multigrid with a standard Gauss-Seidel smoother as well as a new macro Gauss-Seidel smoother. The latter smoother provides good results with respect to both the geometry mapping and the polynomial degree.
