ANALYSIS OF AN XFEM DISCRETIZATION FOR STOKES INTERFACE PROBLEMS

作者：Kirchhart Matthias^*; Gross Sven^*; Reusken Arnold^*

来源：SIAM Journal on Scientific Computing, 2016, 38(2): A1019-A1043.

DOI：10.1137/15M1011779

• Stokes equations
• interface problem
• extended finite element space
• preconditioning
• Schur complement

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 mu(1)/mu(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.

• 出版日期2016