475

IGPM475.pdf January 2018 
TITLE 
A Finite Element Method for the Surface Stokes Problem

AUTHORS 
Maxim A. Olshanskii, Annalisa Quaini, Arnold Reusken, Vladimir Yushutin

ABSTRACT 
We consider a Stokes problem posed on a 2D surface embedded in a 3D domain. The equations describe an equilibrium, areapreserving tangential flow of a viscous surface fluid and serve as a model problem in the dynamics of material interfaces. In this paper, we develop and analyze a Trace finite element method (TraceFEM) for such a surface Stokes problem. TraceFEM relies on finite element spaces defined on a fixed, surfaceindependent background mesh which consists of
shaperegular tetrahedra. Thus, there is no need for surface parametrization or surface fitting with the mesh. The TraceFEM treated here is based on P1 bulk finite elements for both the velocity and the pressure. In order to enforce the velocity vector field to be tangential to the surface we introduce a penalty term. The method is straightforward to implement and has an O(h2) geometric consistency error, which is of the same order as the approximation error due to the P1 –P1 pair for velocity and pressure. We prove stability and optimal order discretization error bounds in the surface H1 and L2 norms. A series of numerical experiments is presented to illustrate certain features of the proposed TraceFEM.

KEYWORDS 
surface fluid equations, surface stokes problems trace finite element method
