A control problem for a linearized time-discrete regularized fracture propagation process is considered. The discretization of the problem is done using a conforming finite element method. In contrast to many works on discretization of PDE constrained optimization problems, the particular setting has to cope with the fact that the linearized fracture equation is not necessarily coercive. A quasi-best approximation result will be shown in the case of an invertible, though not necessarily coercive, linearized fracture equation. Based on this a priori error estimates for the control, state, and adjoint variables will be derived.

中文翻译：

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线性断裂控制问题的先验误差估计

考虑线性化时间离散正则化裂缝扩展过程的控制问题。问题的离散化是使用一致的有限元方法完成的。与许多关于PDE约束优化问题离散化的工作相反，特定设置必须应对线性化裂缝方程不一定具有强制性这一事实。在可逆的但不一定是强制性的线性断裂方程的情况下，将显示一个最佳近似结果。基于此，将得出控制，状态和伴随变量的先验误差估计。