Abstract
We introduce a new finite element (FE) discretization framework applicable
for covariant split equations. The introduction of additional differential
forms (DF) that form pairs with the original ones permits the splitting of the
equations into topological momentum and continuity equations and
metric-dependent closure equations that apply the Hodge-star operator. Our
discretization framework conserves this geometrical structure and provides for
all DFs proper FE spaces such that the differential operators hold in strong
form. We introduce lowest possible order discretizations of the split 1D wave
equations, in which the discrete momentum and continuity equations follow by
trivial projections onto piecewise constant FE spaces, omitting partial
integrations. Approximating the Hodge-star by nontrivial Galerkin projections
(GP), the two discrete metric equations follow by projections onto either the
piecewise 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 relation
with the P1-P1 FE scheme that approximates both variables by piecewise linear
spaces (P1). The split schemes that apply a mixture of GP1 and GP0 share the
dispersion relation with the stable P1-P0 FE scheme that applies piecewise
linear and piecewise constant (P0) spaces. However, the split schemes exhibit
second order convergence for both quantities of interest. For the split scheme
applying twice GP0, we are not aware of a corresponding standard formulation to
compare with. Though it does not provide a satisfactory approximation of the
dispersion relation as short waves are propagated much too fast, the discovery
of the new scheme illustrates the potential of our discretization framework as
a toolbox to study and find FE schemes by new combinations of FE spaces.
