Efficient algorithms for solving the $p$-Laplacian in polynomial time

TL;DR

This paper introduces efficient polynomial-time algorithms based on the barrier method for numerically solving the nonlinear $p$-Laplacian PDE across all parameter values, with confirmed practical performance.

Contribution

It develops a unified barrier method-based algorithm that solves the $p$-Laplacian in $O(\sqrt{n}\log n)$ iterations for all $p$, advancing computational PDE methods.

Findings

Algorithm solves $p$-Laplacian in polynomial time for all $p$

Numerical experiments confirm theoretical iteration bounds

Method is efficient across a wide range of $p$ values

Abstract

The $p$-Laplacian is a nonlinear partial differential equation, parametrized by $p \in [1,\infty]$. We provide new numerical algorithms, based on the barrier method, for solving the $p$-Laplacian numerically in $O(\sqrt{n}\log n)$ Newton iterations for all $p \in [1,\infty]$, where $n$ is the number of grid points. We confirm our estimates with numerical experiments.