论文标题
订单二的不合格有限元方法和三个维度的Stokes流量的第三种方法
Nonconforming finite element methods of order two and order three for the Stokes flow in three dimensions
论文作者
论文摘要
在这项研究中，为Stokes问题构建并利用了第三顺序和第三顺序的不合格有限要素。四面体的所有方面的订单矩最高$ k-1 $（$ k = 2,3 $）用于DOF（自由度），以构建具有$ P_ {k+1} $ expectly的泡泡功能空间的Unisorvent $ k $ strorder-rorder不合格的有限元。事实证明，$ k $ order元素和不连续的零件$ p_ {k} $是稳定的，可以通过保留无元素的无差异状态来解决Stokes问题。建立离散INF-SUP条件的主要困难来自通常无法构建通常的Fortin操作员的事实。得益于气泡功能的明确表示，其差异空间被证明与四面体上的$ p_k $相同，与四面体上的$ p_k $相同，这在克服上述困难方面起着重要作用，并导致了令人期待的离散问题所需的良好范围。此外，使用不连续的分段$ p_ {k-1} $减少了$ k $ - 订单不合格的有限元，并被证明是稳定的，用于解决Stokes问题。缺乏Fortin操作员会导致难以分析减少的三阶元件对的离散INF-SUP条件。为了解决这个问题，采用了所谓的宏观元素技术，具有至关重要的代数结果，即在离散速度空间在宏观元素上相对于离散压力空间的离散速度空间的正交补体空间的函数特性。提供数值实验以验证理论结果。
In this study, the nonconforming finite elements of order two and order three are constructed and exploited for the Stokes problem. The moments of order up to $k-1$ ($k=2,3$) on all the facets of the tetrahedron are used for DoFs (degrees of freedom) to construct the unisolvent $k$-order nonconforming finite element with the bubble function space of $P_{k+1}$ explicitly represented. The pair of the $k$-order element and the discontinuous piecewise $P_{k}$ is proved to be stable for solving the Stokes problem with the element-wise divergence-free condition preserved. The main difficulty in establishing the discrete inf-sup condition comes from the fact that the usual Fortin operator can not be constructed. Thanks to the explicit representation of the bubble functions, its divergence space is proved to be identical to the orthogonal complement space of constants with respect to $P_k$ on the tetrahedron, which plays an important role to overcome the aforementioned difficulty and leads to the desirable well-posedness of the discrete problem. Furthermore, a reduced $k$-order nonconforming finite element with a discontinuous piecewise $P_{k-1}$ is designed and proved to be stable for solving the Stokes problem. The lack of the Fortin operator causes difficulty in analyzing the discrete inf-sup condition for the reduced third-order element pair. To deal with this problem, the so-called macro-element technique is adopted with a crucial algebraic result concerning the property of functions in the orthogonal complement space of the divergence of the discrete velocity space with respect to the discrete pressure space on the macro-element. Numerical experiments are provided to validate the theoretical results.
