Evolving surface finite element methods for random advection-diffusion equations

Abstract

Even though random partial differential equations (PDEs) have become a very active field of research, the analysis and numerical analysis of random PDEs on surfaces still appears to be in its infancy. In this paper, we introduce and analyse a surface finite element discretization of advection-diffusion equations with uncertain coefficients on evolving hypersurfaces. After stating unique solvability of the resulting semi-discrete problem, we prove optimal error bounds for the semi-discrete solution and Monte Carlo samplings of its expectation in appropriate Bochner spaces. Our theoretical findings are illustrated by numerical experiments in two and three space dimensions.