We consider a unique continuation problem where the Dirichlet trace of the solution is known to have finite dimension. We prove Lipschitz stability of the unique continuation problem and design a finite element method that exploits the finite dimensionality to enhance stability. Optimal a priori and a posteriori error estimates are shown for the method. The extension to problems where the trace is not in a finite dimensional space, but can be approximated to high accuracy using finite dimensional functions is discussed. Finally, the theory is illustrated in some numerical examples.

翻译：我们考虑一个唯一延拓问题，其中解的狄利克雷迹已知具有有限维性质。我们证明了该唯一延拓问题的Lipschitz稳定性，并设计了一种利用有限维特性增强稳定性的有限元方法。该方法的最优先验和后验误差估计被给出。进一步讨论了迹不属于有限维空间但可通过有限维函数高精度近似的问题。最后，通过数值算例验证了理论分析。