A FINITE ELEMENT METHOD FOR THE SURFACE STOKES PROBLEM

We consider a Stokes problem posed on a 2D surface embedded in a 3D domain. The equations describe an equilibrium, area-preserving 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. A TraceFEM relies on finite element spaces de fined on a fixed, surface-independent background mesh which consists of shape-regular tetrahedra. Thus, there is no need for surface parametrization or surface fitting with the mesh. The TraceFEM treated here is based on P-1 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(h(2)) geometric consistency error, which is of the same order as the approximation error due to the P-1 P-1 pair for velocity and pressure. We prove stability and optimal order discretization error bounds in the surface H-1 and L-2 norms. A series of numerical experiments is presented to illustrate certain features of the proposed TraceFEM.