Finite element approximations of the stochastic mean curvature flow of planar curves of graphs

TL;DR

This paper introduces finite element methods for approximating the stochastic mean curvature flow of planar curves, providing error estimates and computational experiments to analyze the effects of noise on geometric evolution.

Contribution

It develops and analyzes semi-discrete and fully discrete finite element schemes for a stochastic PDE modeling curve evolution, including regularization and convergence analysis.

Findings

Established strong convergence rates for the proposed methods.

Provided computational experiments illustrating the impact of noise.

Derived error estimates for the regularized SPDE approximation.

Abstract

This paper develops and analyzes a semi-discrete and a fully discrete finite element method for a one-dimensional quasilinear parabolic stochastic partial differential equation (SPDE) which describes the stochastic mean curvature flow for planar curves of graphs. To circumvent the difficulty caused by the low spatial regularity of the SPDE solution, a regularization procedure is first proposed to approximate the SPDE, and an error estimate for the regularized problem is derived. A semi-discrete finite element method, and a space-time fully discrete method are then proposed to approximate the solution of the regularized SPDE problem. Strong convergence with rates are established for both, semi- and fully discrete methods. Computational experiments are provided to study the interplay of the geometric evolution and gradient type-noises.