We present a new structure-preserving numerical method which exhibits high order convergence and does not rely on the geometric realization of any dual mesh. We use B-spline based de Rham complexes to construct two exact sequences of discrete differential forms: the primal sequence starts from the space of tensor-product splines of degree $p$ and at least $C^1$ continuity. Similarly, the dual sequence starts from the space of tensor-product splines of degree $p-1$. The differential operators are condensed in the exterior derivative operator, and due to the high continuity of splines they are well defined both sequences. The method has to be completed with discrete Hodge-star operators that encapsulate all the metric-dependent properties, including material properties, see [1]. We presented in [2] three possible families of discrete Hodge-star operators for splines, which map the space of primal $k$-forms into the space of dual $(n-k)$-forms, and vice versa. For one of those families, we have shown that the operator can be applied through the fast inversion of Kronecker product matrices. This yields computational times much lower than for standard Galerkin discretizations, and the method also exhibits high order of convergence and exact energy conservation [3]. The extension of the method to the multi-patch setting has to face two main issues: the exterior derivative for the dual complex is not well defined, due to the low continuity of the spaces at the interfaces (which is the same as in finite elements); and the dimensions of the spaces between the primal and the dual complex do not match. We will show our progress in applying discontinuous Galerkin techniques to solve these two issues. Bibliography [1] R. Hiptmair, Discrete Hodge operators. Numer. Math., Vol. 90, pp. 265--289, 2001. [2] B. Kapidani and R. Vázquez, High order geometric methods with splines: an analysis of discrete Hodge-star operators. SIAM J. Sci. Comput., Vol. 44, pp. A3673--A3699, 2022. [3] B. Kapidani and R. Vázquez, High order geometric methods with splines: fast solution with explicit time-stepping for {M}axwell equations. J. Comput. Phys., Vol. 493, paper no. 112440, 2023.