A non-conforming composite quadrilateral finite element pair for feedback stabilization of the Stokes equations
A non-conforming composite quadrilateral finite element pair for feedback stabilization of the Stokes equations
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DOI：
10.1515/jnma-2014-0009
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发表时间：
2014-10
影响因子：
3
通讯作者：
P. Benner;J. Saak;F. Schieweck;P. Skrzypacz;H. Weichelt
中科院分区：
文献类型：
--
作者：
P. Benner;J. Saak;F. Schieweck;P. Skrzypacz;H. Weichelt
翻译后摘要-在这方面的贡献，我们展示了一种方法的边界反馈稳定的Stokes问题周围的固定轨迹。我们推导出一个正式的低秩算法，用于解决算子符号的稳定化问题。出现的运营商方程制定的平稳偏微分方程（PDE），而不是使用其有限维表示的矩阵。局部逐点满足发散约束的Galerkin方法特别吸引人，因为它恰当地表示了Helmholtz投影的作用。复合技术的主要优点是单元矩阵的有效组装，使用静态凝聚和对角质量矩阵的计算成本的减少。与协调元相比，复合元的非协调特性保证了更好的稀疏模式，这是由于与相邻单元相对应的基函数之间的耦合数量较少。我们还实现了每个元素的子三角形的逐点质量守恒。
Abstract - In this contribution, we show a method for the boundary feedback stabilization of the Stokes problem around a stationary trajectory. We derive a formal low-rank algorithm for solving the stabilization problem in operator notation. The appearing operator equations are formulated in terms of stationary partial differential equations (PDEs) instead of using their finite dimensional representations in terms of matrices. A Galerkin method, satisfying the divergence constraint pointwise locally is especially appealing since it represents appropriately the action of the Helmholtz projection. The main advantages of the composite technique are the efficient assembly of element matrices, the reduction of computational costs using static condensation, and the diagonal mass matrix. The non-conforming character of the composite element guarantees a better sparsity pattern, compared to conforming elements, due to the lower number of couplings between basis functions corresponding to neighboring cells. We also achieve the pointwise mass conservation on sub-triangles of each element.