Numerical approximation of the first $p$-Laplace eigenpair

TL;DR

This paper develops a numerical scheme to approximate the first eigenpair of the p-Laplace operator for large p values on Euclidean and surface domains, addressing computational challenges and exploring the p→∞ limit.

Contribution

We introduce a surface finite element method combining Newton inverse-power iteration with a domain rescaling strategy for stable large p computations.

Findings

Accurate approximation of the first p-Laplace eigenpair for large p

Demonstrated convergence towards the p→∞ limit in various domains

Validated robustness and accuracy through numerical experiments

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.