Evolving surface finite element methods for random advection-diffusion equations

TL;DR

This paper develops and analyzes evolving surface finite element methods for solving advection-diffusion equations with uncertain coefficients on moving surfaces, providing error bounds and numerical validation.

Contribution

It introduces a novel finite element discretization for advection-diffusion equations on evolving surfaces with uncertainty, and proves optimal error estimates.

Findings

Optimal error bounds for semi-discrete solutions

Validation through numerical experiments in 2D and 3D

Theoretical analysis of Monte Carlo sampling of expectations

Abstract

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.