Differentiation  Elemental Spatial Operators  Integration

Flux Integrals

In the following chapters we will outline the Discontinuous Galerkin Method. In this method it is necessary to evaluate terms of the form: $$\int_{\partial \Omega} f \phi_n + \int_{\Omega} F \phi_n,$$ where $\partial \Omega$ is the boundary of an element $\Omega$, for all the $\phi_n$ test functions in the elemental basis. There are $N\frac{(N+1)}{2}$ test functions for a triangle so the boundary integral is an O(N³) operation. This means that the flux integration is as expensive as the volume integral. We can reduce the cost of this integral by examining the discrete sum form:

where Jⁿ and fⁿ are the Jacobian and flux function for the n^th edge.

We can rewrite the edge₁ flux as:

$$\int_{edge_1} f \phi_n = \sum_{j=0}^{N}\sum_{i=0}^{N} (\frac... ...i)}{w^b_0 }) f^e(a_i) \delta_{j 0} \phi_n(a_i,b_j) w^a_i w^b_j$$   

where:

$$\delta_{ij} = \begin{array}{c} 0 \hspace{4pt} \mbox{if}\hs... ...neq j \ 1 \hspace{4pt} \mbox{if}\hspace{4pt} i=j \end{array}$$ The fluxes for the other edges can be constructed in similar ways. Using this summation representation we can now evaluate the surface flux integral by adding the edge fluxes scaled by weight and Jacobians to the F field and then evaluating one volume integral.
 

10/24/1998