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We introduce a new finite element (FE) discretization framework applicablefor covariant split equations. The introduction of additional differentialforms (DF) that form pairs with the original ones permits the splitting of theequations into topological momentum and continuity equations andmetric-dependent closure equations that apply the Hodge-star operator. Ourdiscretization framework conserves this geometrical structure and provides forall DFs proper FE spaces such that the differential operators hold in strongform. We introduce lowest possible order discretizations of the split 1D waveequations, in which the discrete momentum and continuity equations follow bytrivial projections onto piecewise constant FE spaces, omitting partialintegrations. Approximating the Hodge-star by nontrivial Galerkin projections(GP), the two discrete metric equations follow by projections onto either thepiecewise constant (GP0) or piecewise linear (GP1) space.  Our framework gives us three schemes with significantly different behavior.The split scheme using twice GP1 is unstable and shares the dispersion relationwith the P1-P1 FE scheme that approximates both variables by piecewise linearspaces (P1). The split schemes that apply a mixture of GP1 and GP0 share thedispersion relation with the stable P1-P0 FE scheme that applies piecewiselinear and piecewise constant (P0) spaces. However, the split schemes exhibitsecond order convergence for both quantities of interest. For the split schemeapplying twice GP0, we are not aware of a corresponding standard formulation tocompare with. Though it does not provide a satisfactory approximation of thedispersion relation as short waves are propagated much too fast, the discoveryof the new scheme illustrates the potential of our discretization framework asa toolbox to study and find FE schemes by new combinations of FE spaces.

A structure-preserving split finite element discretization of the split 1D wave equations