A family of C1 quadrilateral finite elements

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∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{\infty }$\end{document}, 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.