Robust Preconditioning for XFEM Applied to Time-Dependent Stokes Problems
Robust Preconditioning for XFEM Applied to Time-Dependent Stokes Problems
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DOI：
10.1137/15m1024007
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发表时间：
2016-11
期刊：
影响因子：
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通讯作者：
S. Gross;Thomas Ludescher;M. Olshanskii;Arnold Reusken
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文献类型：
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作者：
S. Gross;Thomas Ludescher;M. Olshanskii;Arnold Reusken
通过对含时间的Stokes问题进行时间离散化，我们考虑了一个反应项成比例于$\tau=1/\Delta t\geq0$的准定态Stokes界面问题。用于空间离散化的网格未与界面对齐。我们使用$P_1$扩展有限元空间进行压力近似，使用标准协调$P_2$有限元空间进行速度近似。添加了从文献中已知的压力稳定项，因为FE对不是LBB稳定的。对于稳定的离散双线性形式，我们得到了一个新的inf-sup稳定性结果。提出并分析了一种新的Schur补码预处理器。我们给出了一个分析，证明了预处理器关于$h$，$\tau$，其中$\tau\in[0，c_0]\cup[c_1h^{-2}，\inty)$，以及界面位置的鲁棒性。数值结果表明，该预处理器在整个参数范围内是稳健的，对于粘性系数也是稳健的。
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 viscos...