Unique continuation principles for finite-element discretizations of the Laplacian

TL;DR

This paper establishes a discrete version of the unique continuation principle for finite-element discretizations of the Laplacian, demonstrating conditions under which solutions must be zero and providing counterexamples.

Contribution

It proves a novel unique continuation principle for finite-element discretizations of the Laplacian eigenvalue problem, including conditions and counterexamples.

Findings

Unique continuation holds under specific geometric and topological conditions.

Counterexample shows failure when assumptions are not met.

Application to eigenvalue interlacing involving inner solutions.

Abstract

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.