A Surface Finite Element Scheme for a Stochastic PDE on an Evolving Curve

Abstract.

In this paper we consider an evolving surface finite element method 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 finite element method discretization. Our error analysis appropriately generalizes classical estimates to this weaker setting. We present some numerical simulations that confirm our theoretical findings.