In this talk we study discretization spaces over multi-patch domains, which can be used for isogeometric analysis of fourth order partial differential equations (PDEs). While standard Galerkin discretization of fourth order PDE problems require C1-smooth discretizations, we propose a method that uses approximate C1-smoothness, cf. (Weinmüller, Takacs; CMAME, 2021) and (Weinmüller, Takacs; arXiv, 2022).

A key property of IGA is that it is simple to achieve high order smoothness within a single tensor-product B-spline (or NURBS) patch. However, to increase the geometric flexibility, one has to construct spaces beyond such a tensor-product structure. This can be done using unstructured splines, e.g., as in (Takacs, Toshniwal; arXiv, 2022), or using a multi-patch construction. While C0-matching multi-patch domains are easy to construct, C1-smoothness is harder to achieve. It was shown in (Collin, Sangalli, Takacs; CAGD, 2016), that C1-smooth isogeometric discretizations over G1-smooth multi-patch domains do in general not possess sufficient approximation power. This issue was circumvented in (Collin, Sangalli, Takacs; CAGD, 2016) by restricting to a smaller class of G1-smooth multi-patch parametrizations, so called analysis-suitable G1 multi-patch parametrizations, which yield C1-smooth isogeometric spaces. However, to avoid this additional restriction, we relax the smoothness constraints and construct isogeometric spaces that yield optimal convergence rates in numerical experiments while being only approximately C1.

Such constructions are of interest when solving numerically fourth-order problems, such as the biharmonic equation or Kirchhoff-Love plate and shell formulations. The approximate C1 method is advantageous when compared to alternatives that rely on a weak imposition of smoothness, such as Nitsche's method. In contrast to weakly imposing coupling conditions, the approximate C1 construction is explicit and no additional terms need to be introduced to penalize the jump of the derivative at the interface. Thus, the approximate C1 method can be used more easily as no additional parameters need to be estimated.