First-order system least-squares for interface problems

The two-phase flow problem with incompressible flow in the subdomains is studied in this paper. The Stokes flow problems are treated as first-order systems, involving stress and velocity and using the L² norm to define a least-squares functional. A combination of H(div)-conforming Raviart–Thomas and standard H¹-conforming elements is used for the discretization. The interface conditions are directly in the H(div)-conforming finite element space. The homogeneous least-squares functional is shown to be equivalent to an appropriate norm allowing the use of standard finite element approximation estimates. It also establishes the fact that the local evaluation of the least-squares functional itself constitutes an a posteriori error estimator to be used for adaptive refinement strategies.