We consider a stationary Stokes interface problem. In the discretization the interface is not aligned with the triangulation. For the discretization we use the P 1 extended finite element space (P 1-XFEM) for the pressure and the standard conforming P 2 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 µ 1 /µ 2 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 P 1-XFE for pressure and P 2-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.

Abstract. We present a new class of C ∞ -smooth finite element spaces on Cartesian grids, based on a partition of unity approach. We use these spaces to construct smooth approximations of particle fields, i. e., finite sums of weighted Dirac deltas. In order to use the spaces on general domains, we propose a fictitious domain formulation, together with a new high-order accurate stabilization. Stability, convergence, and conservation properties of the scheme are established. Numerical experiments confirm the analysis and show that the Cartesian grid-size σ should be taken proportional to the square-root of the particle spacing h, resulting in significant speed-ups in vortex methods.

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We present a splitting-free variant of the vorticity redistribution method. Spatial consistency and stability when combined with a time-stepping scheme are proven. We propose a new strategy preventing excessive growth in the number of particles while retaining the order of consistency. The novel concept of small neighbourhoods significantly reduces the method's computational cost. In numerical experiments the method showed second order convergence, one order higher than predicted by the analysis. Compared to the fast multipole code used in the velocity computation, the method is about three times faster.