Analysis of an XFEM Discretization for Stokes Interface Problems

We consider a stationary Stokes interface problem. In the discretization the interface is not aligned with the triangulation. For the discretization we use the P1 extended finite element space (P1-XFEM) for the pressure and the standard conforming P2 finite element space for the velocity. Since this pair is not necessarily LBB stable, a consistent stabilization term, known from the literature, is added. For the discrete bilinear form an inf-sup stability result is derived, which is uniform with respect to h (mesh size parameter), the viscosity quotient μ1/μ2 and the position of the interface in the triangulation. Based on this, discretization error bounds are derived. An optimal preconditioner for the stiffness matrix corresponding to this pair P1-XFE for pressure and P2-FE for velocity is presented. The preconditioner has block diagonal form, with a multigrid preconditioner for the velocity block and a new Schur complement preconditioner. Optimality of this block preconditioner is proved. Results of numerical experiments illustrate properties of the discretization method and of a preconditioned MINRES solver. AMS subject classifications. 65N15, 65N22, 65N30, 65F10