A surface finite element scheme for a stochastic PDE on an evolving curve

TL;DR

This paper develops a finite element method for solving a stochastic PDE on a moving curve, addressing challenges from rough stochastic noise and providing error analysis and numerical validation.

Contribution

It introduces a novel surface finite element scheme for stochastic PDEs on evolving curves, including a variational solution concept and error estimates.

Findings

Numerical simulations confirm theoretical error bounds.

The method effectively handles stochastic noise on evolving curves.

Error analysis extends classical estimates to stochastic settings.

Abstract

In this paper we consider an ESFEM method for the advection and diffusion of a scalar quantity on a moving closed curve. The diffusion process is controlled by a forcing term that may include a rough term (specifically a stochastic noise) which in particular destroys the classical time differentiability properties of the solution. We provide a suitable variational solution concept and a fully discrete FEM discretization. Our error analysis appropriately generalizes classical estimates to this weaker setting. We present some numerical simulations that confirm our theoretical findings.