Numerical approximation of the first $p$-Laplace eigenpair

We approximate the first Dirichlet eigenpair of the $p$-Laplace operator for $2\le p<\mathrm{\infty }$ on both Euclidean and surface domains. We emphasize large $p$ values and discuss how the $p\to \mathrm{\infty }$ 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 $p\to \mathrm{\infty }$ limiting behavior.