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In the context of isogeometric analysis, globally $C^1$ isogeometric spacesover unstructured quadrilateral meshes allow the direct solution of fourthorder partial differential equations on complex geometries via their Galerkindiscretization. The design of such smooth spaces has been intensively studiedin the last five years, in particular for the case of planar domains, and isstill task of current research. In this paper, we first give a short survey ofthe developed methods and especially focus on the approach [26]. There, theconstruction of a specific $C^1$ isogeometric spline space for the class ofso-called analysis-suitable $G^1$ multi-patch parametrizations is presented.This particular class of parameterizations comprises exactly those multi-patchgeometries, which ensure the design of $C^1$ spaces with optimal approximationproperties, and allows the representation of complex planar multi-patchdomains. We present known results in a coherent framework, and also extend theconstruction to parametrizations that are not analysis-suitable $G^1$ byallowing higher-degree splines in the neighborhood of the extraordinaryvertices and edges. Finally, we present numerical tests that illustrate thebehavior of the proposed method on representative examples.

Isogeometric analysis with $C^1$ functions on unstructured quadrilateral meshes