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., 2005), which is based on polynomial elements of tensor-product
degree p≥6, to all degrees p≥3. Thus, we call the family of C1
finite elements Brenner-Sung quadrilaterals. The proposed C1 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 C2 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 3×3 or 2×2 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∈{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 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