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Unique continuation principles for finite-element discretizations of the Laplacian

Graham Cox, Scott MacLachlan, Luke Steeves

TL;DR

The paper investigates whether the continuous unique continuation principle for the Laplacian persists under finite-element discretizations on polygonal domains, focusing on $P_1$ and $Q_1$ elements. It establishes a discrete UCP under a geometric angle condition (all interior angles $\le\pi/2$) and a topological boundary zero-forcing set, and it introduces edge-leaky forcing to handle rectangular tensor-product meshes where some off-diagonal entries can be positive. The authors provide counterexamples showing failure of discrete UCP when the mesh fails these conditions, and they show an application to eigenvalue interlacing via a discrete Dirichlet-to-Neumann map, revealing how inner-solution spaces contribute to spectral counts. Overall, the work clarifies how mesh geometry and topology influence the preservation of qualitative PDE properties in FE discretizations and connects these ideas to spectral theory through explicit interlacing relations.

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 $-Δu = λu$ 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.

Unique continuation principles for finite-element discretizations of the Laplacian

TL;DR

The paper investigates whether the continuous unique continuation principle for the Laplacian persists under finite-element discretizations on polygonal domains, focusing on and elements. It establishes a discrete UCP under a geometric angle condition (all interior angles ) and a topological boundary zero-forcing set, and it introduces edge-leaky forcing to handle rectangular tensor-product meshes where some off-diagonal entries can be positive. The authors provide counterexamples showing failure of discrete UCP when the mesh fails these conditions, and they show an application to eigenvalue interlacing via a discrete Dirichlet-to-Neumann map, revealing how inner-solution spaces contribute to spectral counts. Overall, the work clarifies how mesh geometry and topology influence the preservation of qualitative PDE properties in FE discretizations and connects these ideas to spectral theory through explicit interlacing relations.

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 . Namely, we show that any solution to the discretized equation 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.

Paper Structure

This paper contains 20 sections, 10 theorems, 72 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Overview of the finite-element method
  3. The Neumann problem
  4. Interior and boundary degrees of freedom
  5. The Dirichlet problem
  6. The Dirichlet-to-Neumann map
  7. Finite-element spaces
  8. Unique continuation
  9. The graph associated to a mesh
  10. Triangular meshes
  11. Rectangular meshes and leaky forcing
  12. Examples and counterexamples
  13. The regular hexagon
  14. The deformed hexagon
  15. The heptagon
  16. ...and 5 more sections

Key Result

Lemma 3.2

Let $u$ satisfy $Au = \lambda Mu$ for some $\lambda \in \mathbb{R}$. Suppose $u$ vanishes in $N[\boldsymbol x_i] \backslash \{\boldsymbol x_j\}$ for some $j \neq i$. If $a_{i,j} - \lambda m_{i,j} \neq 0$, then $u$ vanishes at $\boldsymbol x_j$.

Figures (6)

  • Figure 1: Two meshes of a hexagonal domain that show the difference between when the nodes on the boundary (highlighted in green) form a zero-forcing set (at left) and do not (at right).
  • Figure 2: At left, a tensor-product mesh on a rectangular domain, and the corresponding graph, with a zero forcing set shown in blue, at right.
  • Figure 3: Triangular mesh for which the $P_1(\Omega)$ discretization satisfies the unique continuation principle via edge-leaky forcing.
  • Figure 4: Meshes of hexagonal and heptagonal domains for which the boundary nodes are not zero-forcing sets.
  • Figure 5: An eigenvector on a multiply-connected domain with vanishing Cauchy data on both boundary components.
  • ...and 1 more figures

Theorems & Definitions (26)

  • Definition 1.1
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Remark 3.4
  • Proposition 3.5
  • proof
  • Remark 3.6
  • ...and 16 more