In this paper, we consider the unique continuation problem for the Schr\"odinger equations. We prove a H\"older type conditional stability estimate and build up a parameterized stabilized finite element scheme adaptive to the \textit{a priori} knowledge of the solution, achieving error estimates in interior domains with convergence up to continuous stability. The approximability of the scheme to solutions with only $H^1$-regularity is studied and the convergence rate for solutions with regularity higher than $H^1$ is also shown. Comparisons in terms of different parameterization for different regularities will be illustrated with respect to the convergence and condition numbers of the linear systems. Finally, numerical experiments will be given to illustrate the theory.

翻译：本文研究薛定谔方程的唯一延拓问题。我们证明了一个Hölder类型的条件稳定性估计，并建立了一种基于解的先验知识自适应的参数化稳定化有限元格式，实现了内域上的误差估计且收敛阶达到连续稳定性。研究了该格式对仅有$H^1$正则性的解的可逼近性，并给出了正则性高于$H^1$的解的收敛速率。针对不同正则性下的不同参数化方式，将在线性系统的收敛性和条件数方面进行对比说明。最后，通过数值实验验证理论结果。