In the first part of this talk, we analyze the convergence of Raviart-Thomas finite elements on curved domains in the context of first-order system least squares formulations. In particular, it will be shown that under sufficient regularity assumptions, the optimal order of convergence is retained on curved domains if a parametric version of Raviart-Thomas elements based on a polynomial mapping is used near the boundary. In particular, we present an estimate for the normal flux on interpolated boundaries.
In the second part, we consider the coupled problem with Stokes flow in two subdomains separated by an interface. At the interface, continuity of the velocity and the momentum balance condition for the stress tensor need to be imposed. The interface is characterized by a level set function which satisfies an appropriate transport equation and the problem can be written as a domain decomposition problem.For numerical results a combination of H(div)-conforming Raviart-Thomas and standard H1-conforming elements is used and an approximation of the momentum balance condition is implemented exactly in the space and controlled with the theory developed in the first part.