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Robust preconditioning for XFEM applied to time-dependent Stokes problems. (English) Zbl 1398.76106

Summary: We consider a quasi-stationary Stokes interface problem with a reaction term proportional to \(\tau=1/\Delta t \geq 0\) 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 \(P_1\) extended finite element space for the pressure approximation and the standard conforming \(P_2\) 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 \(h\), \(\tau\), with \(\tau\in [0,c_0]\cup[c_1 h^{-2},\infty)\), and the position of the interface. Numerical results are included which indicate that the preconditioner is robust for the whole parameter range \(\tau \geq 0\) and also with respect to the viscosity ratio \(\mu_ 1/\mu_2\).

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65F08 Preconditioners for iterative methods
65F10 Iterative numerical methods for linear systems
65N15 Error bounds for boundary value problems involving PDEs
65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
76D07 Stokes and related (Oseen, etc.) flows

Software:

CutFEM
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Full Text: DOI

References:

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