  A new kind of spline geometric method approach is presented. Its main
ingredient is the use of well established spline spaces forming a discrete de
Rham complex to construct a primal sequence $\{X^k_h\}^n_{k=0}$, starting from
splines of degree $p$, and a dual sequence $\{\tilde{X}^k_h\}_{k=0}^n$,
starting from splines of degree $p-1$. By imposing homogeneous boundary
conditions to the spaces of the primal sequence, the two sequences can be
isomorphically mapped into one another. Within this setup, many familiar second
order partial differential equations can be finally accommodated by explicitly
constructing appropriate discrete versions of constitutive relations, called
Hodge--star operators. Several alternatives based on both global and local
projection operators between spline spaces will be proposed. The appeal of the
approach with respect to similar published methods is twofold: firstly, it
exhibits high order convergence. Secondly, it does not rely on the geometric
realization of any (topologically) dual mesh. Several numerical examples in
various space dimensions will be employed to validate the central ideas of the
proposed approach and compare its features with the standard Galerkin approach
in Isogeometric Analysis.
