论文标题
非结构化时空有限元方法用于抛物线方程的最佳控制
Unstructured space-time finite element methods for optimal control of parabolic equations
论文作者
论文摘要
这项工作为抛物线最佳控制问题的数值解决方案提供了完全非结构化的简单时空网格的时空有限元方法。使用Babuška的定理，我们显示了与线性状态方程的典型模型问题的一阶最优系统的适当性，但没有控制约束。这是针对连续和离散级别的。基于这些结果，我们得出离散误差估计。然后，我们考虑由Schlögl模型引起的半连接抛物线最佳控制问题。相关的非线性最优系统是通过牛顿方法来解决的，牛顿的方法在每个牛顿步骤中都必须解决与线性模型问题所考虑的一阶最优系统相似的线性系统。我们提出了各种数值实验，包括基于残留类型误差指标的自适应时空有限元离散的结果。在最后两个示例中，我们还考虑了半连接的抛物线最佳控制问题，并在控件上施加了框限制。
This work presents and analyzes space-time finite element methods on fully unstructured simplicial space-time meshes for the numerical solution of parabolic optimal control problems. Using Babuška's theorem, we show well-posedness of the first-order optimality systems for a typical model problem with linear state equations, but without control constraints. This is done for both continuous and discrete levels. Based on these results, we derive discretization error estimates. Then we consider a semilinear parabolic optimal control problem arising from the Schlögl model. The associated nonlinear optimality system is solved by Newton's method, where a linear system, that is similar to the first-order optimality systems considered for the linear model problems, has to be solved at each Newton step. We present various numerical experiments including results for adaptive space-time finite element discretizations based on residual-type error indicators. In the last two examples, we also consider semilinear parabolic optimal control problems with box constraints imposed on the control.
