A conforming auxiliary space preconditioner for the mass conserving stress-yielding method.

We are studying the efficient solution of the system of linear equations stemming from the mass conserving stress-yielding (MCS) discretization of the Stokes equations. We perform static condensation to arrive at a system for the pressure and velocity unknowns. An auxiliary space preconditioner for the positive definite velocity block makes use of efficient and scalable solvers for conforming Finite Element spaces of low order and is analyzed with emphasis placed on robustness in the polynomial degree of the discretization. Numerical experiments demonstrate the potential of this approach and the efficiency of the implementation.

Divergence-free tangential finite element methods for incompressible flows on surfaces.

In this work we consider the numerical solution of incompressible flows on two-dimensional manifolds. Whereas the compatibility demands of the velocity and the pressure spaces are known from the flat case one further has to deal with the approximation of a velocity field that lies only in the tangential space of the given geometry. Abandoning H 1-conformity allows us to construct finite elements which are-due to an application of the Piola transformation-exactly tangential. To reintroduce continuity (in a weak sense) we make use of (hybrid) discontinuous Galerkin techniques. To further improve this approach, H ( div Γ ) -conforming finite elements can be used to obtain exactly divergence-free velocity solutions. We present several new finite element discretizations. On a number of numerical examples we examine and compare their qualitative properties and accuracy.