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Preprint-No.: <   420   >   Published in: March 2015   PDF-File: IGPM420.pdf
Title:Analysis of an XFEM Discretization for Stokes Interface Problems
Authors:Matthias Kirchhart, Sven Groß, Arnold Reusken
Abstract:
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.
Keywords:Stokes equations, interface problem, extended finite element space, preconditioning, Schur complement
DOI: 10.1137/15m1011779
Publication:SIAM J. Sci. Comput.
38(2), A1019–A1043. (25 pages)
Corresponding author:Sven Gross