A super-smooth <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e1945" altimg="si17.svg"><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math> spline space over planar mixed triangle and quadrilateral meshes

In this paper we introduce a C1 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 et al. (1968), with the C1 quadrilateral element introduced in Brenner and Sung (2005), Kapl et al. (2020) for polynomial degrees p≥5. The space is assumed to be C2 at all vertices and C1 across edges, and the splines are uniquely determined by C2-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 C1 quadrilateral construction, inspired by isogeometric analysis, is two-fold: on one hand, the ability to connect triangle and quadrilateral finite elements in a C1 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 C1 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≥5. The basis is simple to implement and the obtained results are optimal with respect to the mesh size for L∞, L2 as well as Sobolev norms H1 and H2.