A Stabilized Cut Streamline Diffusion Finite Element Method for Convection-Diffusion Problems on Surfaces

TL;DR

This paper introduces a stabilized cut finite element method for solving stationary convection-diffusion problems on surfaces embedded in three-dimensional space, ensuring optimal error estimates and well-conditioned matrices.

Contribution

It develops a novel stabilized cut finite element method with proven optimal error bounds and condition number estimates for surface convection-diffusion problems.

Findings

Optimal order a priori error estimates proved

Condition number of stiffness matrix is O(h^{-1})

Numerical examples confirm theoretical results

Abstract

We develop a stabilized cut finite element method for the stationary convection diffusion problem on a surface embedded in ${\mathbb{R}}^d$. The cut finite element method is based on using an embedding of the surface into a three dimensional mesh consisting of tetrahedra and then using the restriction of the standard piecewise linear continuous elements to a piecewise linear approximation of the surface. The stabilization consists of a standard streamline diffusion stabilization term on the discrete surface and a so called normal gradient stabilization term on the full tetrahedral elements in the active mesh. We prove optimal order a priori error estimates in the standard norm associated with the streamline diffusion method and bounds for the condition number of the resulting stiffness matrix. The condition number is of optimal order $O(h^{-1})$ for a specific choice of method parameters. Numerical example supporting our theoretical results are also included.