We present a novel family of C¹ quadrilateral finite elements, which define global C¹ spaces over a general quadrilateral mesh with vertices of arbitrary valency. The elements extend the construction by Brenner and Sung (J. Sci. Comput. 22(1-3), 83-118, 2005), which is based on polynomial elements of tensor-product degree p ≥ 6, to all degrees p ≥ 3. The proposed C¹ quadrilateral is based upon the construction of multi-patch C¹ isogeometric spaces developed in Kapl et al. (Comput. Aided Geometr. Des. 69, 55–75 2019). The quadrilateral elements possess similar degrees of freedom as the classical Argyris triangles, developed in Argyris et al. (Aeronaut. J. 72(692), 701–709 1968). Just as for the Argyris triangle, we additionally impose C² continuity at the vertices. In contrast to Kapl et al. (Comput. Aided Geometr. Des. 69, 55–75 2019), in this paper, we concentrate on quadrilateral finite elements, which significantly simplifies the construction. We present macro-element constructions, extending the elements in Brenner and Sung (J. Sci. Comput. 22(1–3), 83–118 2005), for polynomial degrees p = 3 and p = 4 by employing a splitting into 3 × 3 or 2 × 2 polynomial pieces, respectively. We moreover provide approximation error bounds in \(L^{\infty }\), L², H¹ and H² for the piecewise-polynomial macro-element constructions of degree p ∈{3,4} and polynomial elements of degree p ≥ 5. Since the elements locally reproduce polynomials of total degree p, the approximation orders are optimal with respect to the mesh size. Note that the proposed construction combines the possibility for spline refinement (equivalent to a regular splitting of quadrilateral finite elements) as in Kapl et al. (Comput. Aided Geometr. Des. 69, 55–75 30) with the purely local description of the finite element space and basis as in Brenner and Sung (J. Sci. Comput. 22(1–3), 83–118 2005). In addition, we describe the construction of a simple, local basis and give for p ∈{3,4,5} explicit formulas for the Bézier or B-spline coefficients of the basis functions. Numerical experiments by solving the biharmonic equation demonstrate the potential of the proposed C¹ quadrilateral finite element for the numerical analysis of fourth order problems, also indicating that (for p = 5) the proposed element performs comparable or in general even better than the Argyris triangle with respect to the number of degrees of freedom.

我们提出了一个新的C ¹四边形有限元族，它定义了具有任意价顶点的一般四边形网格上的全局C ¹空间。这些元素扩展了 Brenner 和 Sung (J. Sci. Comput. 22(1-3), 83-118, 2005) 的构造，该构造基于张量积度p ≥ 6 的多项式元素，扩展到所有度p ≥ 3. 提议的C ¹四边形基于Kapl 等人开发的多面片C ^{1 等}几何空间的构造。（计算辅助几何。Des. 69, 55–75 2019)。四边形单元具有与 Argyris 等人开发的经典 Argyris 三角形相似的自由度。（Aeronaut. J. 72 (692), 701–709 1968）。与 Argyris 三角形一样，我们还在顶点处附加了C ²连续性。与 Kapl 等人相反。(Comput. Aided Geometr. Des. 69 , 55–75 2019)，在本文中，我们专注于四边形有限元，这大大简化了构造。我们提出宏元素构造，扩展 Brenner 和 Sung (J. Sci. Comput. 22 (1-3), 83-118 2005) 中的元素，用于多项式次数p = 3 和p= 4 通过分别拆分为 3 × 3 或 2 × 2 多项式。而且我们提供近似误差界限在\（L ^ {\ infty} \） ，大号²，ħ ¹和ħ ²为度的分段多项式宏观元件结构p ∈{3,4}和度的多项式元素p ≥ 5. 由于元素局部再现总次数为p 的多项式，因此近似阶数对于网格大小是最佳的。请注意，所提出的构造结合了 Kapl 等人中的样条细化的可能性（相当于四边形有限元的规则分裂）。（计算。辅助几何。Des。69 , 55–75 30) 和 Brenner 和 Sung (J. Sci. Comput. 22 (1–3), 83–118 2005)中对有限元空间和基的纯局部描述。此外，我们描述了一个简单的局部基的构造，并为p ∈{3,4,5} 给出了基函数的 Bézier 或 B 样条系数的显式公式。通过求解双调和方程的数值实验证明了所提出的C ¹四边形有限元在四阶问题的数值分析中的潜力，也表明（对于p = 5）所提出的单元的性能与 Argyris 三角形相当甚至更好关于自由度的数量。