Master element defined in local coordinates ξ and η showing the degrees of freedom hi the
value of h at the midpoint of edge i opposite of corner node i.
Basis and degrees of freedom
As emphasized by Cockburn and Shu (1998), we note here that a judicious choice of
basis functions can simplify the implementation of the scheme and improve the computational
efficiency. Owing to the fact that discontinuities are permitted across element interfaces, the
choice of the basis functions are not limited by the requirement of continuity as in the CG finite
element method. Therefore, one can choose degrees of freedom that, for example, save cost in
evaluating the integrals in Eq. (15) and/or simplify the implementation of the slope limiter. In our
implementation, we use piecewise linear triangular elements described below.
Considering the "master element" as shown in Fig. 3 defined in the transformed
coordinates ξ and η, the approximate solution zh can be expressed as:
zh = ∑ zi (t )φi (ξ ,η )