A family of C1C^1 quadrilateral finite elements

We present a novel family of $C^1$ quadrilateral finite elements, which define global $C^1$ spaces over a general quadrilateral mesh with vertices of arbitrary valency. The elements extend the construction by (Brenner and Sung, J. Sci. Comput., 2005), which is based on polynomial elements of tensor-product degree $p\geq 6$, to all degrees $p \geq 3$. Thus, we call the family of $C^1$ finite elements Brenner-Sung quadrilaterals. The proposed $C^1$ quadrilateral can be seen as a special case of the Argyris isogeometric element of (Kapl, Sangalli and Takacs, CAGD, 2019). The quadrilateral elements possess similar degrees of freedom as the classical Argyris triangles. Just as for the Argyris triangle, we additionally impose $C^2$ continuity at the vertices. In this paper we focus on the lower degree cases, that may be desirable for their lower computational cost and better conditioning of the basis: We consider indeed the polynomial quadrilateral of (bi-)degree~$5$, and the polynomial degrees $p=3$ and $p=4$ by employing a splitting into $3\times3$ or $2\times2$ polynomial pieces, respectively. The proposed elements reproduce polynomials of total degree $p$. We show that the space provides optimal approximation order. Due to the interpolation properties, the error bounds are local on each element. In addition, we describe the construction of a simple, local basis and give for $p\in\{3,4,5\}$ explicit formulas for the B\'{e}zier or B-spline coefficients of the basis functions. Numerical experiments by solving the biharmonic equation demonstrate the potential of the proposed $C^1$ 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

A family of C-1 quadrilateral finite elements

We present a novel family of C-1 quadrilateral finite elements, which define global C-1 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 C-1 quadrilateral is based upon the construction of multi-patch C-1 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 C-2 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 x 3 or 2 x 2 polynomial pieces, respectively. We moreover provide approximation error bounds in L-infinity, L-2, H-1 and H-2 for the piecewise-polynomial macro-element constructions of degree p is an element of{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 is an element of{3,4,5} explicit formulas for the Bezier 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