Analysis and Approximations of Dirichlet Boundary Control of Stokes Flows in the Energy Space
Analysis and Approximations of Dirichlet Boundary Control of Stokes Flows in the Energy Space
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DOI：
10.1137/21m1406799
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发表时间：
2020-11
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影响因子：
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通讯作者：
W. Gong;M. Mateos;J. Singler;Yangwen Zhang
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文献类型：
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作者：
W. Gong;M. Mateos;J. Singler;Yangwen Zhang
研究了二维多边形区域中Stokes流的Dirichlet边界控制问题。我们考虑具有两个不同边界控制正则化项的代价泛函：L^2范数和能量空间范数。我们证明了这两个问题的适定性和规律性的结果，这两个问题的有限元离散，并证明有限元误差估计后一个问题。研究能量空间问题的动机来自于我们的分析：我们证明了控制空间${\bm L}^2（\Gamma）$的选择可以导致在角点处具有不连续性的最优控制，即使域是凸的。我们在数值实验中观察到这种现象。这种行为不会发生在Dirichlet边界控制问题的Poisson方程的凸多边形区域，并可能是不可取的真实的应用。对于能量空间问题，我们得到了一阶最优性条件，并证明了控制问题的解比正则化问题的解更正则.我们还证明了先验误差估计的控制的能量范数，并提出了几个数值实验的控制问题的凸和非凸域。
We study Dirichlet boundary control of Stokes flows in 2D polygonal domains. We consider cost functionals with two different boundary control regularization terms: the $L^2$ norm and an energy space seminorm. We prove well-posedness and regularity results for both problems, develop finite element discretizations for both problems, and prove finite element error estimates for the latter problem. The motivation to study the energy space problem follows from our analysis: we prove that the choice of the control space ${\bm L}^2(\Gamma)$ can lead to an optimal control with discontinuities at the corners, even when the domain is convex. We observe this phenomenon in numerical experiments. This behavior does not occur in Dirichlet boundary control problems for the Poisson equation on convex polygonal domains, and may not be desirable in real applications. For the energy space problem, we derive the first order optimality conditions, and show that the solution of the control problem is more regular than the solution of the problem with the ${\bm L}^2(\Gamma)$ regularization. We also prove a priori error estimates for the control in the energy norm, and present several numerical experiments for both control problems on convex and nonconvex domains.