耦合半线性PDE-ODE系统非光滑最优控制问题时空有限元逼近的先验误差估计
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
摘要：
本文研究了简化半线性梯度增强损伤模型的时空Galerkin有限元离散的先验误差估计。模型方程是一个特殊的结构，状态方程由一个椭圆型偏微分方程，它必须在几乎所有的时间都得到满足，再加上一个非光滑，半线性常微分方程，必须在几乎所有的空间点成立。时间上采用常值间断Galerkin法离散，空间上采用协调线性有限元离散。对于不可控方程，我们证明了时间上的线性收敛性和空间上的离散化误差的$\mathcal{O}(h^{\frac{3}{2}-\varepsilon})$阶。我们关于最优控制问题的主要结果是dG（0）cG（1）-离散控制一致收敛性于$l\in H^1_{\{0\}}(0,T;L^2(Ω))$ .控制的误差估计通过二次增长条件建立。数值实验证明了收敛速度。
英文摘要：
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.
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