Unique continuation principles are fundamental properties of elliptic partial differential equations, giving conditions that guarantee that the solution to an elliptic equation must be uniformly zero. Since finite-element discretizations are a natural tool to help gain understanding into elliptic equations, it is natural to ask if such principles also hold at the discrete level. In this work, we prove a version of the unique continuation principle for piecewise-linear and -bilinear finite-element discretizations of the Laplacian eigenvalue problem on polygonal domains in $\mathbb{R}^2$. Namely, we show that any solution to the discretized equation $-\Delta u = \lambda u$ with vanishing Dirichlet and Neumann traces must be identically zero under certain geometric and topological assumptions on the resulting triangulation. We also provide a counterexample, showing that a nonzero \emph{inner solution} exists when the topological assumptions are not satisfied. Finally, we give an application to an eigenvalue interlacing problem, where the space of inner solutions makes an explicit appearance.

翻译：唯一延拓原理是椭圆型偏微分方程的基本性质，它给出了保证椭圆型方程解必须一致为零的条件。由于有限元离散化是理解椭圆型方程的自然工具，人们自然会问此类原理在离散层面是否同样成立。本文针对$\mathbb{R}^2$多边形区域上拉普拉斯特征值问题的分段线性与双线性有限元离散化，证明了唯一延拓原理的一个版本。具体而言，我们证明在所得三角剖分满足特定几何与拓扑假设的条件下，离散方程$-\Delta u = \lambda u$的任何具有消失狄利克雷迹与诺伊曼迹的解必恒为零。同时我们构造了一个反例，表明当拓扑假设不满足时存在非零的\emph{内解}。最后，我们将结果应用于特征值交错问题，其中内解空间将显式出现。