正方形和立方体上三次Serendipity空间的Hermite和Bernstein型基函数
Hermite and Bernstein Style Basis Functions for Cubic Serendipity Spaces on Squares and Cubes
AI总结：
本文为三次Serendipity有限元空间引入了新的Hermite型和Bernstein型几何分解，这些空间维度更小但仍能保证三次收敛阶，且基函数与自由度及几何有规范关系，适用于等几何分析。
摘要：
我们为三次Serendipity有限元空间$S_3(I^2)$和$S_3(I^3)$引入了新的Hermite型和Bernstein型几何分解，这些空间定义在Arnold和Awanou的最新工作[Found. Comput. Math. 11 (2011), 337--344]中。Serendipity空间的维度远小于常用的双三次和三三次Hermite张量积空间——正方形上为12而非16，立方体上为32而非64——但在有限元方法中使用时，仍能保证在$H^1$范数下获得三次阶 extit{先验}误差估计。我们定义的基函数与有限元自由度以及其图形的几何之间具有规范关系；这意味着这些基函数可能适用于等几何分析应用，其中域几何和域上支持的函数由相同的基函数描述。此外，这些基函数是常用的双三次和三三次多项式Bernstein或Hermite基函数的线性组合，从而可以快速融入现有的有限元代码中。
英文摘要：
We introduce new Hermite-style and Bernstein-style geometric decompositions of the cubic order serendipity finite element spaces $S_3(I^2)$ and $S_3(I^3)$, as defined in the recent work of Arnold and Awanou [Found. Comput. Math. 11 (2011), 337--344]. The serendipity spaces are substantially smaller in dimension than the more commonly used bicubic and tricubic Hermite tensor product spaces - 12 instead of 16 for the square and 32 instead of 64 for the cube - yet are still guaranteed to obtain cubic order \textit{a priori} error estimates in $H^1$ norm when used in finite element methods. The basis functions we define have a canonical relationship both to the finite element degrees of freedom as well as to the geometry of their graphs; this means the bases may be suitable for applications employing isogeometric analysis where domain geometry and functions supported on the domain are described by the same basis functions. Moreover, the basis functions are linear combinations of the commonly used bicubic and tricubic polynomial Bernstein or Hermite basis functions, allowing their rapid incorporation into existing finite element codes.
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