Analysis of an XFEM Discretization for Stokes Interface Problems
Analysis of an XFEM Discretization for Stokes Interface Problems
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DOI：
10.1137/15m1011779
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发表时间：
2016-03
期刊：
影响因子：
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通讯作者：
Matthias Kirchhart;S. Gross;Arnold Reusken
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文献类型：
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作者：
Matthias Kirchhart;S. Gross;Arnold Reusken
我们考虑平稳斯托克斯接口问题。在离散化中，界面不与三角测量对齐。对于离散化，我们使用 $P_1$ 扩展有限元空间 ($P_1$-XFEM) 表示压力，使用符合标准的 $P_2$ 有限元空间表示速度。由于该对不一定是 LBB 稳定的，因此添加了从文献中已知的一致稳定项。对于离散双线性形式，导出了 inf-sup 稳定性结果，该结果对于 $h$（网格尺寸参数）、粘性商 $\mu_1/\mu_2$ 以及三角剖分中界面的位置是一致的。在此基础上，推导出离散化误差界。提出了对应于这对压力 $P_1$-XFE 和速度 $P_2$-FE 的刚度矩阵的最佳预处理器。预处理器具有块对角形式，具有用于速度块的多重网格预处理器和新的 Schur 补预处理器。该块预处理器的最优性...
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 ...