We consider a stable unique continuation problem for the wave equation where the initial data is lacking and the solution is reconstructed using measurements in some subset of the bulk domain. Typically fairly sophisticated space-time methods have been used in previous work to obtain stable and accurate solutions to this reconstruction problem. Here we propose to solve the problem using a standard discontinuous Galerkin method for the temporal discretization and continuous finite elements for the space discretization. Error estimates are established under a geometric control condition. We also investigate two preconditioning strategies which can be used to solve the arising globally coupled space-time system by means of simple time-stepping procedures. Our numerical experiments test the performance of these strategies and highlight the importance of the geometric control condition for reconstructing the solution beyond the data domain.

翻译：我们考虑波动方程的一类稳定唯一延拓问题，该问题中初始数据缺失，需利用体域内某子集上的测量值重构解。以往工作中，通常采用较为复杂的时空耦合法来获得该重构问题的稳定精确解。本文提出采用标准间断伽辽金法进行时间离散、连续有限元法进行空间离散来求解该问题。在几何控制条件下建立了误差估计。此外，我们研究了两种预处理策略，可通过简单的时间推进过程求解由此产生的全局耦合时空系统。数值实验验证了这些策略的性能，并凸显了几何控制条件在数据域外重构解中的重要性。