In this talk we introduce a new structure-preserving Galerkin Differences (GD) formulation grounded on continuous and discrete de Rham complexes constructed within the framework of Discrete Exterior Calculus (DEC). Similarly to Finite-Volume (FV) or Finite-Difference (FD) methods, high-order spatial accuracy is achieved without introducing additional interior degrees of freedom. For each geometric degree of freedom, compactly supported piecewise-polynomial basis functions are defined by means of sliding local polynomial spaces. These spaces are constructed locally at the cell level by enforcing only the required compatibility conditions. In multidimensional Cartesian geometries, the resulting discrete spaces are obtained through a tensor-product construction. As a consequence, the numerical solution inside each cell is, by design, a high-order polynomial reconstructed from the local degrees of freedom. We then present a higher-order accurate realization of mimetic Galerkin Differences applied to the compressible magnetohydrodynamics (MHD) equations. The proposed spatial discretization combines a hybrid Finite Volume (FV) and Galerkin Differences (GD) approach. In a first implementation, the nonlinear convection–acoustic contributions are handled by a robust high-order FV WENO scheme, whereas the magnetic terms are discretized using the proposed mimetic Galerkin Differences formulation. The use of DEC allows several geometric properties to be preserved at the discrete level.