arXiv subjects

Genming Bai

Publications and source records attributed to Genming Bai.

7 recordsLinked to original sources

arXiv2026

Stabilization and optimal $L^2$ convergence of Dziuk's method with piecewise linear parametric finite elements for curve-shortening flow

We propose a stabilized version of the fully discrete Dziuk's method for the curve-shortening flow of a closed planar curve with piecewise linear parametric finite elements. With a carefully designed stabilization term, we are able to show a surprising discrete tangential stability of the Barrett--Garcke--N\"urnberg (BGN) type under the parabolic scaling $\tau\simeq h^2$---a feature hidden at the continuous level. Together with a new super-approximation result for the reversely averaged normal vector of linear elements, this Dziuk-type discrete tangential stability yields optimal $L^2$ convergence.

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arXiv2026

A spherical harmonic pseudo-spectral approach to mean curvature flow of surfaces with spherical topology

We propose and analyze a spherical harmonic pseudo-spectral method for the mean curvature flow of closed surfaces with spherical topology. The evolving surface is represented by a global parametrization over the unit sphere, and the continuous weak formulation underlying Dziuk's method [G. Dziuk, Numer. Math., 1991] is discretized in a finite-dimensional space of spherical harmonics. By explicitly taking into account quadrature errors on the autonomously evolving numerical surface, we prove exponential convergence of the position error under the assumption that the initial global parametrization is analytic. The convergence analysis developed herein is general and could apply to other moving-domain and geometric evolution problems. Numerical experiments confirm the theoretical result.

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arXiv2025

A convergent finite element method for two-phase Stokes flow driven by surface tension

We present the first convergence proof for an iso-parametric finite element discretization of two-phase Stokes flow in $\Omega \subset \mathbb{R}^d$, $d=2,3$, with interface dynamics governed by mean curvature. The proof relies on a crucial discrete coupled parabolicity structure of the error system and a powerful iso-parametric framework of convergence analysis where we do not really discriminate consistency and stability. This new mixing idea leads to a non-trivial construction of the bulk mesh in the consistency analysis. The techniques and analysis developed in this paper provide fundamental numerical analysis tools for general curvature-driven free boundary problems.

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arXiv2025

Convergence analysis for the Barrett--Garcke--Nurnberg method of transport type for evolving curves

In this paper, we propose a Barrett-Garcke-Nurnberg (BGN) method for evolving geometries under general flows and present the corresponding convergence analysis. Specifically, we examine the scenario where a closed curve evolves according to a prescribed background velocity field. Unlike mean curvature flow and surface diffusion, where the evolution velocities inherently exhibit parabolicity, this case is dominated by transport which poses a significant difficulty in establishing convergence proofs. To address the challenges imposed by this transport-dominant nature, we derive several discrete energy estimates of the transport type on discretized polynomial surfaces within the framework of the projection error. The use of the projection error is indispensable as it provides crucial additional stability through its orthogonality structure. We prove that the proposed method converges sub-optimally in the L2 norm, and this is the first convergence proof for a fully discrete numerical method solving the evolution of curves driven by general flows.

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arXiv2024

Maximal regularity of evolving FEMs for parabolic equations on an evolving surface

In this paper, we prove that spatially semi-discrete evolving finite element method for parabolic equations on a given evolving hypersurface of arbitrary dimensions preserves the maximal $L^p$-regularity at the discrete level. We first establish the results on a stationary surface and then extend them, via a perturbation argument, to the case where the underlying surface is evolving under a prescribed velocity field. The proof combines techniques in evolving finite element method, properties of Green's functions on (discretised) closed surfaces, and local energy estimates for finite element methods

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arXiv2024

Weak maximum principle of finite element methods for parabolic equations in polygonal domains

The weak maximum principle of finite element methods for parabolic equations is proved for both semi-discretization in space and fully discrete methods with $k$-step backward differentiation formulae for $k = 1,... ,6$, on a two-dimensional general polygonal domain or a three-dimensional convex polyhedral domain. The semi-discrete result is established via a dyadic decomposition argument and local energy estimates in which the nonsmoothness of the domain can be handled. The fully discrete result for multistep backward differentiation formulae is proved by utilizing the solution representation via the discrete Laplace transform and the resolvent estimates, which are inspired by the analysis of convolutional quadrature for parabolic and fractional-order partial differential equations.

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arXiv2022

Physics Informed Neural Networks (PINNs)for approximating nonlinear dispersive PDEs

We propose a novel algorithm, based on physics-informed neural networks (PINNs) to efficiently approximate solutions of nonlinear dispersive PDEs such as the KdV-Kawahara, Camassa-Holm and Benjamin-Ono equations. The stability of solutions of these dispersive PDEs is leveraged to prove rigorous bounds on the resulting error. We present several numerical experiments to demonstrate that PINNs can approximate solutions of these dispersive PDEs very accurately

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