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Numerical approximation of the first $p$-Laplace eigenpair

Hannah Potgieter, Razvan C. Fetecau, Steven J. Ruuth

TL;DR

This paper develops a robust numerical framework for the first Dirichlet p-Laplacian eigenpair on both Euclidean and surface domains, emphasizing large p. It combines a surface finite element method with a Newton inverse-power iteration and introduces a domain rescaling and p-continuation strategy to stabilize computations up to p ≈ 100. The approach is validated against exact 1D solutions and explored on disks, squares, hemispheres, half-tori, and complex surface geometries, demonstrating convergence toward the distance-to-boundary in the p → ∞ limit and revealing geometry-dependent limiting behavior. The work advances surface PDE eigenproblem computation and provides tools for analyzing domain geometry through p-Laplacian eigenpairs, with potential applications in shape optimization and geometry processing.

Abstract

We approximate the first Dirichlet eigenpair of the $p$-Laplace operator for $2 \leq p < \infty$ on both Euclidean and surface domains. We emphasize large $p$ values and discuss how the $p \to \infty$ limit connects to the underlying geometry of our domain. Working with large $p$ values introduces significant numerical challenges. We present a surface finite element numerical scheme that combines a Newton inverse-power iteration with a new domain rescaling strategy, which enables stable computations for large $p$. Numerical experiments in $1$D, planar domains, and surfaces embedded in $\mathbb{R}^3$ demonstrate the accuracy and robustness of our approach and show convergence towards the $p \to \infty$ limiting behavior.

Numerical approximation of the first $p$-Laplace eigenpair

TL;DR

This paper develops a robust numerical framework for the first Dirichlet p-Laplacian eigenpair on both Euclidean and surface domains, emphasizing large p. It combines a surface finite element method with a Newton inverse-power iteration and introduces a domain rescaling and p-continuation strategy to stabilize computations up to p ≈ 100. The approach is validated against exact 1D solutions and explored on disks, squares, hemispheres, half-tori, and complex surface geometries, demonstrating convergence toward the distance-to-boundary in the p → ∞ limit and revealing geometry-dependent limiting behavior. The work advances surface PDE eigenproblem computation and provides tools for analyzing domain geometry through p-Laplacian eigenpairs, with potential applications in shape optimization and geometry processing.

Abstract

We approximate the first Dirichlet eigenpair of the -Laplace operator for on both Euclidean and surface domains. We emphasize large values and discuss how the limit connects to the underlying geometry of our domain. Working with large values introduces significant numerical challenges. We present a surface finite element numerical scheme that combines a Newton inverse-power iteration with a new domain rescaling strategy, which enables stable computations for large . Numerical experiments in D, planar domains, and surfaces embedded in demonstrate the accuracy and robustness of our approach and show convergence towards the limiting behavior.

Paper Structure

This paper contains 20 sections, 1 theorem, 75 equations, 13 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Dirichlet $p$-Laplace eigenvalue problem
  4. The first eigenpair
  5. Asymptotic behavior
  6. Analytic solution in $1$D
  7. Numerical scheme
  8. Handling large $p$ values
  9. Computational results
  10. One–dimensional case $\Omega = \{ x \in \mathbb{R} : a < x < b \}$
  11. Euclidean planar domains
  12. Disk $\Omega = \{ (x, y) \in \mathbb{R}^2 : \sqrt{x^2+y^2} < R$}
  13. Square $\Omega = \{ (x, y) \in \mathbb{R}^2 : |x|+|y| < c$}
  14. Surface domains
  15. Hemisphere $\Omega = \{ (x, y, z) \in \mathbb{R}^3 : \sqrt{x^2+y^2+z^2} = R, \; z > 0$}
  16. ...and 5 more sections

Key Result

Proposition B.2

Let $\Omega$ be an open and bounded subset of a surface $S$, and assume that the set $\mathcal{M}$ of maximal distance-to-boundary points coincides exactly with the ridge set $\mathcal{R}$ (see eqn:calR and eqn:calM). Then the function $d({\bf x}) = \mathrm{dist}({\bf x}, \partial \Omega)$ is a posi where $\Lambda_{\infty} = \|d\|_\infty^{-1}$.

Figures (13)

  • Figure 1: Left: $u_{p}(x) = \sin_p(\frac{\pi_p}{2}(x+1))$ converges to $u_{\infty}(x) = 1-|x|$ to as $p$ increases. Right: pointwise errors between $u_{p}(x)$ and $u_{\infty}(x)$.
  • Figure 2: Illustration of the rescaling transformation $T_\alpha$. On the left, the original rectangular domain contains a largest inscribed circle of radius $R = 1/\alpha$. After applying $T_\alpha$, the domain is scaled so that the maximal distance to the boundary becomes $R = 1$.
  • Figure 3: Illustration of the maximal distance approximation $d_m$ on a hand surface mesh. The marked point (orange) corresponds to the location where the distance to the boundary $\partial \Omega$ (blue) is maximized, giving $d_m \approx 1.956$. This value is used in the scaling $\alpha = d_m^{-1}$.
  • Figure 4: Left: Eigenvalue approximations on some optimally scaled domains for $2 \leq p \leq 100$ shown in log-log scale. Right: Corresponding $p$th roots of eigenvalue approximations and limit $\Lambda_{\infty}$. The disk and hemisphere meshes consist of $327{,}680$ cells, the square mesh consists of $262{,}144$ cells, the half torus mesh consists of $1{,}572{,}864$ cells, and the $1$D mesh consists of $2{,}048$ cells.
  • Figure 5: Contour plots of the first eigenfunction $u_{p, h}$ on the disk. Plots (a)–(d) correspond to different values of $p$, and plot (e) shows the limiting solution as $p \to \infty$. All computations use a mesh with $327{,}680$ cells. Isolines are shown at the same levels across plots.
  • ...and 8 more figures

Theorems & Definitions (5)

  • Remark 3.1
  • Definition B.1: Viscosity solution
  • Proposition B.2
  • proof
  • Remark B.3