arXiv subjects

Hannah Potgieter

Publications and source records attributed to Hannah Potgieter.

4 recordsLinked to original sources

arXiv2026

Deep-Learning Solvers and Surrogates for Infinity and p-Laplace Problems

We investigate the use of neural network solvers for infinity and $p$-Laplace problems, which are fundamental in nonlinear analysis and have practical applications. Our approach employs Physics-Informed Neural Networks (PINNs) and Deep Operator Networks (DeepONets) to address computational challenges associated with large $p$ values, ranging from $2$ to $1000$, on various 2D and 3D domains. Our method offers advantages over traditional physics-based solvers, especially in three dimensions where mesh-based solvers become very costly for these problems. We also establish conditional convergence results for PINN approximations of both problems and a universal approximation result for DeepONet on the parametric $p$-Poisson problem. We demonstrate the effectiveness of these neural network solvers through numerical experiments and compare their performance with conventional methods.

math.NA↗

arXiv2026

Numerical approximation of the first $p$-Laplace eigenpair

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.

math.NA↗

arXiv2025

Geodesic distance approximation using a surface finite element method for the $p$-Laplacian

We use the $p$-Laplacian with large $p$-values in order to approximate geodesic distances to features on surfaces. This differs from Fayolle and Belyaev's (2018) [1] computational results using the $p$-Laplacian for the distance-to-surface problem. Our approach appears to offer some distinct advantages over other popular PDE-based distance function approximation methods. We employ a surface finite element scheme and demonstrate numerical convergence to the true geodesic distance functions. We check that our numerical results adhere to the triangle inequality and examine robustness against geometric noise such as vertex perturbations. We also present comparisons of our method with the heat method from Crane et al. [2] and the classical polyhedral method from Mitchell et al. [3].

cs.GR↗

arXiv2022

Modeling the Second Harmonic in Surface Water Waves Using Generalizations of NLS

When a mechanical wavemaker at one end of a water-wave tank oscillates with a frequency, $ω_0$, time series of downstream surface waves typically include the dominant frequency (or first harmonic), $ω_0$, along with the second, $2ω_0$; third, $3ω_0$; and higher harmonics. This behavior is common for the propagation of weakly nonlinear waves with a narrow band of frequencies centered around the dominant frequency such as in the evolution of ocean swell, pulse propagation in optical fibers, and Langmuir waves in plasmas. Presented herein are measurements of the amplitudes of the first and second harmonic bands from four surface water wave laboratory experiments. The Stokes expansion for small-amplitude surface water waves provides predictions for the amplitudes of the second and higher harmonics given the amplitude of the first harmonic. Similarly, the derivations of the NLS equation and its generalizations (models for the evolution of weakly nonlinear, narrow-banded waves) provide predictions for the second and third harmonic bands given measurements of the first harmonic band. We test the accuracy of these predictions by making two types of comparisons with the experimental measurements. First, we consider the evolution of the second harmonic band while neglecting all other harmonic bands. Second, we use explicit Stokes and generalized NLS formulas to predict the evolution of the second harmonic band using the first harmonic data as input. Comparisons of both types show reasonable agreement, though predictions obtained from dissipative generalizations of NLS consistently outperform the conservative ones. Finally, we show that the predictions obtained from these two methods are qualitatively different.

physics.flu-dyn↗