论文标题
由耦合的半线性PDE系统控制的非平滑最佳控制问题的时空有限元近似的先验误差估计值
A priori error estimates for the space-time finite element approximation of a non-smooth optimal control problem governed by a coupled semilinear PDE-ODE system
论文作者
论文摘要
在本文中，我们研究了简化半线性梯度增强损伤模型的时空盖尔金有限元离散化的先验误差估计。该模型方程是特殊的结构，因为状态方程由椭圆形PDE组成，该椭圆形几乎必须在任何时候都必须与非平滑的半线性颂歌相结合，该颂歌几乎必须在太空中的所有点中保持真实。该系统在时间上通过恒定的不连续的Galerkin方法离散，并且通常在空间中符合线性有限元。对于不受控制的方程式，我们证明了时间的线性收敛，$ \ MATHCAL {O}（H^{\ frac {3} {2} {2} - \ varepsilon}）$用于太空中的离散错误。关于最佳控制问题的我们的主要结果是DG（0）Cg（1）-Discrete Controls to $ L \ in H^1 _ {\ {0 \}}（0，T; l^2（ω））$的均匀收敛。对控件的错误估计是通过二次生长条件建立的。添加数值实验以说明促成的收敛速率。
In this paper we investigate a priori error estimates for the space-time Galerkin finite element discretization of a simplified semilinear gradient enhanced damage model. The model equations are of a special structure as the state equation consists of an elliptic PDE which has to be fulfilled at almost all times coupled with a non-smooth, semilinear ODE that has to hold true in almost all points in space. The system is discretized by a constant discontinuous Galerkin method in time and usual conforming linear finite elements in space. For the uncontrolled equation, we prove linear convergence in time and an order of $\mathcal{O}(h^{\frac{3}{2}-\varepsilon})$ for the discretization error in space. Our main result regarding the optimal control problem is the uniform convergence of dG(0)cG(1)-discrete controls to $l\in H^1_{\{0\}}(0,T;L^2(Ω))$. Error estimates for the controls are established via a quadratic growth condition. Numerical experiments are added to illustrate the proven rates of convergence.
