We consider data assimilation for the heat equation using a finite element space semi-discretization. The approach is optimization based, but the design of regularization operators and parameters rely on techniques from the theory of stabilized finite elements. The space semi-discretized system is shown to admit a unique solution. Combining sharp estimates of the numerical stability of the discrete scheme and conditional stability estimates of the ill-posed continuous pde-model we then derive error estimates that reflect the approximation order of the finite element space and the stability of the continuous model. Two different data assimilation situations with different stability properties are considered to illustrate the framework. Full detail on how to adapt known stability estimates for the continuous model to work with the numerical analysis framework is given in “Appendix”.

中文翻译：

---

使用稳定有限元方法对热方程进行数据同化

我们考虑使用有限元空间半离散化的热方程的数据同化。该方法基于优化，但正则化算子和参数的设计依赖于来自稳定有限元理论的技术。空间半离散化系统被证明承认一个独特的解决方案。结合离散方案的数值稳定性的尖锐估计和不适定连续 pde 模型的条件稳定性估计，我们得出反映有限元空间近似阶数和连续模型稳定性的误差估计。考虑了具有不同稳定性属性的两种不同数据同化情况来说明该框架。