Robust Preconditioning for XFEM Applied to Time-Dependent Stokes Problems

We consider a quasi-stationary Stokes interface problem with a reaction term proportional to τ = 1/∆t ≥ 0 as obtained by a time discretization of a time-dependent Stokes problem. The mesh used for space discretization is not aligned with the interface. We use the P1 extended finite element space for the pressure approximation and the standard conforming P2 finite element space for the velocity approximation. A pressure stabilization term known from the literature is added, since the FE pair is not LBB stable. For the stabilized discrete bilinear form we derive a new inf-sup stability result. A new Schur complement preconditioner is proposed and analyzed. We present an analysis which proves robustness of the preconditioner with respect to h, τ , with τ ∈ [0, c0] ∪ [c1h−2,∞), and the position of the interface. Numerical results are included which indicate that the preconditioner is robust for the whole parameter range τ ≥ 0 and also with respect to the viscosity ratio μ1/μ2.