Isogeometric analysis asked for spline based discretizations of the de Rham complex. This was done in 2011 for tensor product splines [1] , but the extensions to hierarchical splines (a space that allow for local refinement) was not realized till 2020 in 2D [2] and 2024 in higher dimension [3]. Contrary to FEM, compatible discretizations of the de Rham complex using adaptive splines spaces require some mesh and refinement constraints.
Another drawback of hierarchical (and other adaptive) spline spaces is that by break the tesnor product structure they increase the algorithmic complexity and disallow optimized algorithms such as sum-factorization. A possible way around this problem is to use an overlapping space decomposition where each space is tensor product enabling the use of efficient algorithms at the expense of dealing with linear dependency, singular systems and of an increase in the number of degrees of freedom [4].
This talk is about compatible discretizations based on overlapping tensor product spline discretizations.
[1] Buffa, A., Rivas, J., Sangalli, G., & Vázquez, R. (2011). Isogeometric discrete differential forms in three dimensions. SIAM Journal on Numerical Analysis, 49(2), 818-844.
[2] Evans, J. A., Scott, M. A., Shepherd, K. M., Thomas, D. C., & Vázquez Hernández, R. (2020). Hierarchical B-spline complexes of discrete differential forms. IMA Journal of Numerical Analysis, 40(1), 422-473.
[3] Shepherd, K., & Toshniwal, D. (2024). Locally-Verifiable Sufficient Conditions for Exactness of the Hierarchical B-spline Discrete de Rham Complex in R n. Foundations of Computational Mathematics, 1-43.
[4] Bressan, A., Martinelli, M., & Sangalli, G. (2026) Overlapping Subspaces and Singular Systems with Application to Isogeometric Analysis. Domain Decomposition Methods in Science and Engineering XXVIII, https://www.ddm.org/DD28/Proc-28.php