where the degrees of freedom, zi are the values of the approximate solution at the mid-point of
each edge and the basis functions, φi define the linear element of Crouzeix and Raviart (1973)
which for the master element shown in Fig. 3 can be written in the form:
φ1 = 1 - 2ξ ,
φ2 = 1 - 2η ,
φ3 = 2ξ + 2η - 1
(18)
There are several things to note about this basis. The functions φi are equal to 1 at the mid-point
of each edge i and 0 at the mid-points of the other two edges. The basis functions are orthogonal
over an element, specifically:
⎧1/ 6, i = j
∫
φiφ j dξ dη = ⎨
(19)
i≠ j
⎩0,
Ωm
where Ωm denotes the domain of the master element. This property, of course, gives rise to an
orthogonal mass matrix that can be trivially inverted.
Lastly, in the continuous projection
procedure to be described, we will make use of the value of the approximate solution at the
vertices of the triangle. The value of zh at vertex i, denoted by zvi, which is the vertex opposite of
edge i (see Fig. 3), is easily computed as:
3
zvi = -2 zi + ∑ z j
(20)
j =1
As a final note, we remark that the orthogonal, hierarchical, "modal" type basis proposed
by Dubiner (1991), which simplifies p refinement and also adaptivity, can easily be implemented
within the framework of the DG method.
14
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