Local Projection Stabilization for the Stokes System on Anisotropic Quadrilateral Meshes
Local Projection Stabilization for the Stokes System on Anisotropic Quadrilateral Meshes
复制标题
各向异性四边形网格上斯托克斯系统的局部投影稳定
DOI：
10.1007/978-3-540-34288-5_75
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发表时间：
2006
期刊：
影响因子：
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通讯作者：
T. Richter
中科院分区：
文献类型：
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作者：
M. Braack;T. Richter
当采用等阶有限元和N-S对流项时，局部投影稳定化方法适用于稳定Stokes系统的鞍点结构。因此，它已经成功地应用于计算流体力学的不同领域，如三维不可压缩流动[7]、可压缩流动[12]、反应流动[8]、参数估计[4，5]和最优控制问题[11]。虽然在所有这些应用中，这种稳定技术都使用了局部细化网格，但到目前为止，网格一直是各向同性的。各向异性网格上的偏微分方程解对于求解具有内层或边界层的问题是非常重要的，例如在高雷诺数的流体力学中。众所周知，稳定的有限元格式，例如流线迎风Petrov-Galerkin(SUPG)，见[10]，或压力稳定Petrov-Galerkin(PSPG，见[9])，必须在各向异性的情况下进行修改。Becker在[2]中展示了如何在各向异性笛卡尔网格上修改PSPG稳定化。在这项工作中，我们通过在区域Ω⊂R中考虑速度v和压力p的Stokes系统，在各向异性四边形网格上建立了LPS的第一步：
The local projection stabilization (LPS) is suitable to stabilize the saddle point structure of the Stokes system when equal-order finite elements are used, as well as convective terms for Navier-Stokes. Hence, it has already been applied with large success to different fields of computational fluid dynamics, e.g., in 3D incompressible flows [7], compressible flows [12], reactive flows [8], parameter estimation [4, 5] and optimal control problems [11]. Although locally refined meshes have been used for this stabilization technique in all of these applications, the meshes have been isotropic so far. The solution of partial differential equations on anisotropic meshes are of substantial importance for efficient solutions of problems with interior layers or boundary layers, as for instance in fluid dynamics at higher Reynolds number. It is well known that stabilized finite element schemes, e.g. streamline upwind Petrov-Galerkin (SUPG), see [10], or pressure stabilized Petrov-Galerkin (PSPG), see [9], must be modified in the case of anisotropy. Becker has shown in [2] how the PSPG stabilization should be modified on anisotropic Cartesian grids. In this work, we make the first step of formulating LPS on anisotropic quadrilateral meshes by considering the Stokes system in the domain Ω ⊂ R for velocity v and pressure p: