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\mytitle{The Argyris isogeometric space on unstructured
  multi-patch planar domains}

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\author{
\begin{tabular}{c}
 Giancarlo Sangalli\\ %
   Dipartimento di Matematica ``F. Casorati'', Universit\`a degli Studi di Pavia, Italy
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}
\newcommand{\W}{\mathcal{A}} % Argyris space 


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Multi-patch spline parametrizations are used in geometric design and isogeometric
  analysis to represent complex domains. 
We deal with a particular class of $C^0$  planar multi-patch spline parametrizations called analysis-suitable $G^1$ (AS-$G^{1}$) multi-patch parametrizations (cf. \cite{CoSaTa16}). 
This class of parametrizations has to satisfy specific geometric
continuity constraints, and is of importance since it allows to
construct, on the multi-patch domain, $C^1$ isogeometric spaces with optimal approximation properties. It was demonstrated in \cite{KaSaTa17b} that AS-$G^1$ multi-patch parametrizations 
are suitable for modeling 
complex planar multi-patch domains. 

In this work, we present a basis, and an associated dual basis, for
a specific $C^1$ isogeometric spline space~$\W$ over a
given AS-$G^1$ multi-patch parametrization. { We call the space $\W$ the Argyris isogeometric space, since it is $C^1$ across interfaces and $C^2$ at all vertices and generalizes the 
idea of Argyris finite elements (see \cite{argyris1968tuba}) to tensor-product splines.} The considered space~$\W$ is a subspace of the entire $C^1$ isogeometric space~$\mathcal{V}^{1}$, 
which maintains the reproduction properties of traces and normal derivatives along the interfaces. { Moreover, it reproduces all derivatives up to second order at the vertices.}
In contrast to $\mathcal{V}^{1}$, the dimension of 
$\W$  does not depend on the domain parametrization, and
$\W$  admits a  basis and dual
basis which  possess a simple explicit 
representation and   local support. 

{ We conclude with some numerical experiments, which exhibit the optimal approximation order of the Argyris isogeometric space $\W$ and demonstrate the applicability of our approach for isogeometric analysis.}

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\paragraph{Joint work with:} Mario Kapl (Johann Radon Institute for
Computational and Applied Mathematics, Austrian Academy of Sciences,
Austria), Thomas Takacs (Institute of Applied Geometry, Johannes Kepler University Linz, Austria).


%%%% BIBLIOGRAPHY -- MODIFY THIS %%%%%
\begin{thebibliography}{1}


\bibitem{argyris1968tuba}
J.~H. Argyris, I.~Fried, and D.~W. Scharpf.
\newblock The {TUBA} family of plate elements for the matrix displacement
  method.
\newblock {\em The Aeronautical Journal}, 72(692):701--709, 1968.

\bibitem{BeMa14}
M.~Bercovier and T.~Matskewich.
\newblock {\em Smooth {B}\'{e}zier Surfaces over Unstructured Quadrilateral
  Meshes}.
\newblock Lecture Notes of the Unione Matematica Italiana, Springer, 2017.

\bibitem{CoSaTa16}
A.~Collin, G.~Sangalli, and T.~Takacs.
\newblock Analysis-suitable {G}$^1$ multi-patch parametrizations for {C}$^1$
  isogeometric spaces.
\newblock {\em Computer Aided Geometric Design}, 47:93 -- 113, 2016.

\bibitem{KaBuBeJu16}
M.~Kapl, F.~Buchegger, M.~Bercovier, and B.~J\"uttler.
\newblock Isogeometric analysis with geometrically continuous functions on
  planar multi-patch geometries.
\newblock {\em Comput. Methods Appl. Mech. Engrg.}, 316:209 -- 234, 2017.

\bibitem{KaSaTa17}
M.~Kapl, G.~Sangalli, and T.~Takacs.
\newblock Dimension and basis construction for analysis-suitable {G}$^{1}$
  two-patch parameterizations.
\newblock {\em Computer Aided Geometric Design}, 52--53:75 -- 89, 2017.

\bibitem{KaSaTa17b}
M.~Kapl, G.~Sangalli, and T.~Takacs.
\newblock Construction of analysis-suitable {G}$^{1}$ planar multi-patch
  parameterizations.
\newblock {\em Computer-Aided Design}, 97:41 -- 55, 2018.

\bibitem{KaViJu15}
M.~Kapl, V.~Vitrih, B.~J\"uttler, and K.~Birner.
\newblock Isogeometric analysis with geometrically continuous functions on
  two-patch geometries.
\newblock {\em Comput. Math. Appl.}, 70(7):1518 -- 1538, 2015.

\end{thebibliography}





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