A question about Pure Convection solved by Discontinuous-Galerkin FEM

General Discussion
1

The code in documentation is:

problem Adual(cc, w)
    = int2d(Th)(
          (cc/dt+(v1*dx(cc)+v2*dy(cc)))*w
    )
    + intalledges(Th)(
          (1-nTonEdge)*w*(al*abs(n)-n/2)*jump(cc)
    )
    - int2d(Th)(
          ccold*w/dt
    );

1 - nTonEdge = 0 where in the set of boundary edges

and = -1 where in the set of inner edges

So, the formulation should be

\sum_{T \in T_h} \int_T \left( \frac{c^{n+1} - c^n}{\delta t} + u \cdot \nabla c\right) w + \sum_{T \in T_h} \int_{\partial T \cap E} (-1)\left( \alpha | u \cdot n| - \frac{1}{2}u \cdot n \right)[c^{n+1}]w = 0,

where E is the set of inner edges. (E_{\Gamma}^{-} is the set of boundary edges where u \cdot n < 0 and in that case there is no such edges)

But when I read the artical to learn the so-called dual-P_1^{DC} formulation
(see [ERN2006]
ERN, A. and GUERMOND, J. L. Discontinuous Galerkin methods for Friedrichs’ symmetric systems.I. General theory. SIAM J. Numer. Anal.)
It shows:

979×212 19.6 KB

,
q2
q21081×511 87.5 KB

and
q3
q31062×706 122 KB

so from (17) and (27) I got (ignore the term of E_{\Gamma}^{-})

\sum_{T \in T_h} \int_T \left( \frac{c^{n+1} - c^n}{\delta t} + u \cdot \nabla c\right) w + \sum_{T \in T_h} \int_{\partial T \cap E} \left( S_F - \frac{1}{2} \mathcal{D} \right)[c^{n+1}]w = 0,

I’m not sure where the mistake I make that caused the loss of the −1 corresponding to 1−nTonEdge. (And the program is correct)

2

This is because in Ern, Guermond, the jump is taken as (internal)-(external), wherease in FreeFem it is the converse.

1 Like