NDLI: Robust Preconditioning for XFEM Applied to Time-Dependent Stokes Problems

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Reusken, Arnold Olshanskii, Maxim Gross, Sven Ludescher, Thomas
Copyright Year	2016
Abstract	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 $\mu_ 1/\mu_2$.
Sponsorship	Deutsche Forschungsgemeinschaft. Division of Mathematical Sciences
Starting Page	A3492
Ending Page	A3514
Page Count	23
File Format	PDF
ISSN	10648275
DOI	10.1137/15M1024007
e-ISSN	10957197
Issue Number	6
Volume Number	38
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2016-11-01
Access Restriction	Subscribed
Subject Keyword	extended finite elements Solution of discretized equations stabilization Iterative methods for linear systems Error bounds Schur complement preconditioning Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods Stokes equations interface problem
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Computational Mathematics

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