where the degrees of freedom, *z*i 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 *z*h at vertex *i*, denoted by *z*vi, which is the vertex opposite of

edge *i *(see Fig. 3), is easily computed as:

3

*zvi *= -2 *z*i + ∑ 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