  In this paper we introduce a $C^1$ spline space over mixed meshes composed of
triangles and quadrilaterals, suitable for FEM-based or isogeometric analysis.
In this context, a mesh is considered to be a partition of a planar polygonal
domain into triangles and/or quadrilaterals. The proposed space combines the
Argyris triangle, cf. (Argyris, Fried, Scharpf; 1968), with the $C^1$
quadrilateral element introduced in (Brenner, Sung; 2005) and (Kapl, Sangalli,
Takacs; 2019) for polynomial degrees $p\geq 5$. The space is assumed to be
$C^2$ at all vertices and $C^1$ across edges, and the splines are uniquely
determined by $C^2$-data at the vertices, values and normal derivatives at
chosen points on the edges, and values at some additional points in the
interior of the elements.
  The motivation for combining the Argyris triangle element with a recent $C^1$
quadrilateral construction, inspired by isogeometric analysis, is two-fold: on
one hand, the ability to connect triangle and quadrilateral finite elements in
a $C^1$ fashion is non-trivial and of theoretical interest. We provide not only
approximation error bounds but also numerical tests verifying the results. On
the other hand, the construction facilitates the meshing process by allowing
more flexibility while remaining $C^1$ everywhere. This is for instance
relevant when trimming of tensor-product B-splines is performed.
  In the presented construction we assume to have (bi)linear element mappings
and piecewise polynomial function spaces of arbitrary degree $p\geq 5$. The
basis is simple to implement and the obtained results are optimal with respect
to the mesh size for $L^\infty$, $L^2$ as well as Sobolev norms $H^1$ and
$H^2$.
