We study the space of \(C^{2}\)-smooth isogeometric functions on bilinearly parameterized multi-patch domains \(\Omega \subset \mathbb{R}^{2}\), where the graph of each isogeometric function is a ...multi-patch spline surface of bidegree \((d,d)\), \(d \in \{5,6 \}\). The space is fully characterized by the equivalence of the \(C^2\)-smoothness of an isogeometric function and the \(G^2\)-smoothness of its graph surface, cf. (Groisser and Peters,2015; Kapl et al.,2015). This is the reason to call its functions \(C^{2}\)-smooth geometrically continuous isogeometric functions. In particular, the dimension of this \(C^{2}\)-smooth isogeometric space is investigated. The study is based on the decomposition of the space into three subspaces and is an extension of the work (Kapl and Vitrih, 2017) to the multi-patch case. In addition, we present an algorithm for the construction of a basis, and use the resulting globally \(C^{2}\)-smooth functions for numerical experiments, such as performing \(L^{2}\) approximation and solving triharmonic equation, on bilinear multi-patch domains. The numerical results indicate optimal approximation order.
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 \(H^2\).
