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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 <jats:italic>p<\/jats:italic> \u2265\u20096, to all degrees <jats:italic>p<\/jats:italic> \u2265\u20093. The proposed <jats:italic>C<\/jats:italic><jats:sup>1<\/jats:sup> quadrilateral is based upon the construction of multi-patch <jats:italic>C<\/jats:italic><jats:sup>1<\/jats:sup> isogeometric spaces developed in Kapl et al. (Comput. Aided Geometr. Des. <jats:bold>69<\/jats:bold>, 55\u201375 2019). The quadrilateral elements possess similar degrees of freedom as the classical Argyris triangles, developed in Argyris et al. (Aeronaut. J. <jats:bold>72<\/jats:bold>(692), 701\u2013709 1968). Just as for the Argyris triangle, we additionally impose <jats:italic>C<\/jats:italic><jats:sup>2<\/jats:sup> continuity at the vertices. In contrast to Kapl et al. (Comput. Aided Geometr. Des. <jats:bold>69<\/jats:bold>, 55\u201375 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. <jats:bold>22<\/jats:bold>(1\u20133), 83\u2013118 2005), for polynomial degrees <jats:italic>p<\/jats:italic> =\u20093 and <jats:italic>p<\/jats:italic> =\u20094 by employing a splitting into 3 \u00d7 3 or 2 \u00d7 2 polynomial pieces, respectively. We moreover provide approximation error bounds in <jats:inline-formula><jats:alternatives><jats:tex-math>$L^{\\infty }$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mrow>\n                      <mml:mi>L<\/mml:mi>\n                    <\/mml:mrow>\n                    <mml:mrow>\n                      <mml:mi>\u221e<\/mml:mi>\n                    <\/mml:mrow>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, <jats:italic>L<\/jats:italic><jats:sup>2<\/jats:sup>, <jats:italic>H<\/jats:italic><jats:sup>1<\/jats:sup> and <jats:italic>H<\/jats:italic><jats:sup>2<\/jats:sup> for the piecewise-polynomial macro-element constructions of degree <jats:italic>p<\/jats:italic> \u2208{3,4} and polynomial elements of degree <jats:italic>p<\/jats:italic> \u2265\u20095. Since the elements locally reproduce polynomials of total degree <jats:italic>p<\/jats:italic>, 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. <jats:bold>69<\/jats:bold>, 55\u201375 30) with the purely local description of the finite element space and basis as in Brenner and Sung (J. Sci. Comput. <jats:bold>22<\/jats:bold>(1\u20133), 83\u2013118 2005). In addition, we describe the construction of a simple, local basis and give for <jats:italic>p<\/jats:italic> \u2208{3,4,5} explicit formulas for the B\u00e9zier or B-spline coefficients of the basis functions. Numerical experiments by solving the biharmonic equation demonstrate the potential of the proposed <jats:italic>C<\/jats:italic><jats:sup>1<\/jats:sup> quadrilateral finite element for the numerical analysis of fourth order problems, also indicating that (for <jats:italic>p<\/jats:italic> =\u20095) the proposed element performs comparable or in general even better than the Argyris triangle with respect to the number of degrees of freedom.<\/jats:p>","DOI":"10.1007\/s10444-021-09878-3","type":"journal-article","created":{"date-parts":[[2021,11,5]],"date-time":"2021-11-05T14:09:26Z","timestamp":1636121366000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":19,"title":["A family of C1 quadrilateral finite elements"],"prefix":"10.1007","volume":"47","author":[{"given":"Mario","family":"Kapl","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Giancarlo","family":"Sangalli","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9335-4577","authenticated-orcid":false,"given":"Thomas","family":"Takacs","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2021,11,3]]},"reference":[{"issue":"692","key":"9878_CR1","doi-asserted-by":"publisher","first-page":"701","DOI":"10.1017\/S000192400008489X","volume":"72","author":"JH Argyris","year":"1968","unstructured":"Argyris, J. 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