For example, for 1D Burgers equation
$$ u u_x = 0 \\ $$
equivalently,
$$ \frac{dF(u)}{dx} = 0\\ F(u)= \frac{u^2}{2} $$
If I want to obtain $A_{ij},i\ne j$ for two DOFs ($U_i$ and $U_j$) of two elements ($E_0$ and $E_1$) sharing an internal interface $\Gamma_0 $.
I need to substitute $u =\phi_i$ and $v=\phi_j$ to nonlinear-linear form
$$\begin{aligned}
a(u;v) &= \int_{\Omega}{\nabla F(u)\cdot \nabla v dx} - \int_{\Gamma_0}{[F(u) v]dS} \\&=\int_{\Omega}{\nabla F(u)\cdot \nabla v dx} - \int_{\Gamma_0}{\tilde F(u^+,u^-) [v]dS}
\end{aligned}
$$
I understand what $F(u)$ or $\tilde F(u^+,u^-)$ are in the sense of fluid mechanics and conservation laws. But I am really confusing what $F(\phi_i)$ or $\tilde F(\phi_i^+,\phi_j^-)$ is in the sense of physics and how it is integrated. They are so unnatural.
I can read the program codes, but they are all using certain iterations over elements rather than iteration over basis functions. They are indeed correct but I am not sure they are correct for certain $A_{ij},i\ne j$.
