A stabilized finite element method for advection-diffusion equations on surfaces

TL;DR

This paper presents a stabilized finite element method using SUPG for solving advection-diffusion equations on surfaces, addressing stability issues when advection dominates diffusion, with proven error bounds and numerical validation.

Contribution

It introduces a SUPG-based stabilization technique for surface advection-diffusion equations, improving stability and accuracy over existing methods.

Findings

The stabilized method remains stable for coarse meshes.

Numerical experiments confirm improved accuracy.

Error analysis supports theoretical stability claims.

Abstract

A recently developed Eulerian finite element method is applied to solve advection-diffusion equations posed on hypersurfaces. When transport processes on a surface dominate over diffusion, finite element methods tend to be unstable unless the mesh is sufficiently fine. The paper introduces a stabilized finite element formulation based on the SUPG technique. An error analysis of the method is given. Results of numerical experiments are presented that illustrate the performance of the stabilized method.