تقرير
A new error analysis for parabolic Dirichlet boundary control problems
| العنوان: | A new error analysis for parabolic Dirichlet boundary control problems |
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| المؤلفون: | Liang, Dongdong, Gong, Wei, Xie, Xiaoping |
| سنة النشر: | 2023 |
| المجموعة: | Computer Science Mathematics |
| مصطلحات موضوعية: | Mathematics - Numerical Analysis, Mathematics - Optimization and Control, 49J20, 65N15, 65N30 |
| الوصف: | In this paper, we consider the finite element approximation to a parabolic Dirichlet boundary control problem and establish new a priori error estimates. In the temporal semi-discretization we apply the DG(0) method for the state and the variational discretization for the control, and obtain the convergence rates $O(k^{\frac{1}{4}})$ and $O(k^{\frac{3}{4}-\varepsilon})$ $(\varepsilon>0)$ for the control for problems posed on polytopes with $y_0\in L^2(\Omega)$, $y_d\in L^2(I;L^2(\Omega))$ and smooth domains with $y_0\in H^{\frac{1}{2}}(\Omega)$, $y_d\in L^2(I;H^1(\Omega))\cap H^{\frac{1}{2}}(I;L^2(\Omega))$, respectively. In the fully discretization of the optimal control problem posed on polytopal domains, we apply the DG(0)-CG(1) method for the state and the variational discretization approach for the control, and derive the convergence order $O(k^{\frac{1}{4}} +h^{\frac{1}{2}})$, which improves the known results by removing the mesh size condition $k=O(h^2)$ between the space mesh size $h$ and the time step $k$. As a byproduct, we obtain a priori error estimate $O(h+k^{1\over 2})$ for the fully discretization of parabolic equations with inhomogeneous Dirichlet data posed on polytopes, which also improves the known error estimate by removing the above mesh size condition. |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2306.15911 |
| رقم الأكسشن: | edsarx.2306.15911 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |