Stability and error analysis for a diffuse interface approach to an advection-diffusion equation on a moving surface

TL;DR

This paper provides a stability and error analysis for a diffuse interface numerical scheme solving a parabolic PDE on a moving surface, supported by theoretical bounds and numerical tests.

Contribution

It introduces a fully discrete scheme using a level set-based diffuse interface approach with proven stability and error bounds for moving surface PDEs.

Findings

Stability bounds depend on grid size, time step, and interface width.

Error estimates are established under certain conditions.

Numerical tests confirm the theoretical analysis.

Abstract

In this paper we analyze a fully discrete numerical scheme for solving a parabolic PDE on a moving surface. The method is based on a diffuse interface approach that involves a level set description of the moving surface. Under suitable conditions on the spatial grid size, the time step and the interface width we obtain stability and error bounds with respect to natural norms. Furthermore, we present test calculations that confirm our analysis.