Studies of fracture mechanics via phase-field modeling has gained great interest due to the possibility of capturing complex fractures path and branching and coalescence phenomena without a-priori knowledge on the fracture evolution. However, the applicability of such methodology is limited in real-world scenarios due to the entailed computational burden. High-order phase-field models [1], solved in primal form via isogeometric analysis, limit such an effort enabling coarse discretizations while maintaining a comparable accuracy. To further reduce the computational burden, local mesh-adaptivity can be considered as a viable approach. In a recent paper [2], the fourth-order Cahn-Hilliard model is simulated refining the mesh near the phase interface while the discretization is coarsened in the remaining regions. To improve the accuracy of the adopted numerical scheme, mesh-admissibility for THB-splines is explored. In this work, we pose the foundations to combine the advantages of mesh-adaptivity and high-order phase-field models for fracture. Note to the organizers: Presentation in pairs delivered by Luigi Greco and Michele Torre for EYIC MINISYMPOSIUM. REFERENCES [1] L. Greco, A. Patton, M. Negri, A. Marengo, U. Perego, A. Reali. Higher order phase-field modeling of brittle fracture via Isogeometric Analysis. To appear on Engineering with Computers, 2024. [2] C. Bracco, C. Giannelli, A. Reali, M. Torre, R. V´azquez. Adaptive isogeometric phase-field modeling of the Cahn–Hilliard equation: Suitably graded hierarchical refinement and coarsening on multi-patch geometries. Computer Methods in Applied Mechanics and Engineering, 417, 116355, 2023.