469

IGPM469.pdf August 2017 
TITLE 
A Trace Finite Element Method for VectorLaplacians on Surfaces 
AUTHORS 
Sven Groß, Thomas Jankuhn, Maxim A. Olshanskii, Arnold Reusken 
ABSTRACT 
We consider a vectorLaplace problem posed on a 2D surface embedded in a 3D domain, which results from the modeling of surface fluids based on exterior Cartesian differential operators. The main topic of this paper is the development and analysis of a finite element method for the discretization of this surface partial differential equation. We apply the trace finite element technique, in which finite element spaces on a background shaperegular tetrahedral mesh that is surfaceindependent are used for discretization. In order to satisfy the constraint that the solution vector field is tangential to the surface we introduce a Lagrange multiplier. We show wellposedness of the resulting saddle point formulation. A discrete variant of this formulation is introduced which contains suitable stabilization terms and is based on trace finite element spaces. For this method we derive optimal discretization error bounds. Furthermore algebraic properties of the resulting discrete saddle point problem are studied. In particular an optimal Schur complement preconditioner is proposed. Results of a numerical experiment are included. 
KEYWORDS 
surface fluid equations, surface vectorLaplacian, trace finite element method 