A penalty finite element method for a fluid system posed on embedded surface

The paper introduces a finite element method for the incompressible Navier--Stokes equations posed on a closed surface $\Gamma\subset\R^3$. The method needs a shape regular tetrahedra mesh in $\mathbb{R}^3$ to discretize equations on the surface, which can cut through this mesh in a fairly arbitrary way. Stability and error analysis of the fully discrete (in space and in time) scheme is given. The tangentiality condition for the velocity field on $\Gamma$ is enforced weakly by a penalty term. The paper studies both theoretically and numerically the dependence of the error on the penalty parameter. Several numerical examples demonstrate convergence and conservation properties of the finite element method.