While the interest in higher-order models in physics and mechanics grows, their numerical simulation still poses a challenge, especially for arbitrary shaped three-dimensional domains. This contribution presents the mathematical framework as well as the application to different problems in the field of material science, fracture mechanics and diffusion problems. All models under consideration require at least C 1 continuity, which prevents the application of standard finite element analysis and local mesh refinements. Introducing isogeometric analysis (IGA) for the discretization in a finite element framework enables us to deal with these requirements. Moreover, a general hierarchical refinement scheme based on a subdivision projection is presented here for one, two and three dimensional simulations. This technique allows to enhance the approximation space using finer splines on each level but preserves the partition of unity as well as the continuity properties of the original discretization. Using this mathematical framework, the improved convergence of a Kuramoto–Sivashinsky model, a mesh-adapted thermal diffusion simulation and computations of a priori unknown crack propagation in different fracture modes underline the versatility of the presented hierarchical refinement scheme. 
Isogeometric analysis and hierarchical refinement for higher-order phase-field models