  In this article we study a mixed finite element formulation for solving the
Stokes problem with general surface forces that induce a jump of the normal
trace of the stress tensor, on an interface that splits the domain into two
subdomains. Equality of velocities is assumed at the interface. The interface
conditions are taken into account with multipliers. A suitable Lagrangian
functional is introduced for deriving a weak formulation of the problem. A
specificity of this work is the consideration of the interface with a
fictitious domain approach. The latter is inspired by the XFEM approach in the
sense that cut-off functions are used, but it is simpler to implement since no
enrichment by singular functions is provided. In that context, getting
convergence for the dual variables defined on the interface is non-trivial. For
that purpose, an augmented Lagrangian technique stabilizes the convergence of
the multipliers, which is important because their value would determine the
dynamics of the interface in an unsteady framework. Theoretical analysis is
provided, where we show that a discrete inf-sup condition, independent of the
mesh size, is satisfied for the stabilized formulation. This guarantees optimal
convergence rates, that we observe with numerical tests. The capacity of the
method is demonstrated with robustness tests, and with an unsteady model tested
for deformations of the interface that correspond to ellipsoidal shapes in
dimension 2.
