I am a little confused about where does the mass and stiff matrix come from. In Discontinuous Galerkin we divide the domain in elements, $\Omega = \cup^K_{k=1} D^k$. Then assume the solution $u$ can be locally approximated by a polynomial as
$$u_h=\sum_{i=1}^{Np}\hat u_i^k \Phi_i$$
We then introduce the approximated solution into the conservation law to get the residual
$$\mathcal{R} = \frac{\partial u_h}{\partial t} + \frac{\partial au_h}{\partial x} = 0$$
We define now a space of test functions $V_h = \cup^K_{k=1}V_h^k$ where a locally defined member of $V_h^k$ is defined as
$$\phi_h^k=\sum_{j=1}^{N_p} \hat \phi_j^k \Phi_j$$
Finally we require that the residual to be orthogonal to all the test functions in $V_h$ yielding
$$\int_{D^k}\mathcal{R}(x,t)\Phi_j dx= 0$$
- Why do we say that the residual must be orthogonal to all test functions? Shouldn't we need it to be just orthogonal to the functions locally?
- In the inner product between the residual and the test functions, why are we using $\Phi_j$ instead of $\phi_h^k$? Is it because we are asking to be orthogonal to all the test functions, i.e. the basis of the test functions? So we don't care about the coefficients $\hat \phi_j^k$ ?
- How does the mass/stiffness matrix appear? This is just notation probably, but I don't see how from the integral a matrix is built.
EDIT
Perhaps I didn't explain myself correctly. So, given the PDE we multiply by a test function and integrate by parts over spatial domain, gives
$$\int_{D^k} \frac{\partial u_h}{\partial t} \phi_j+ \frac{\partial au_h}{\partial x} \phi_j dx$$ $$\int_{D^k} \frac{\partial u_h}{\partial t} \phi_j+ \frac{\partial \phi_j}{\partial x} au_h = -[(au_h)\phi_j]^{k+1}_{k}$$
Now suppose $u_h = \sum_{i=1}^{N_p}\hat u_h \phi_i$
I get something like
$$\int_{D^k} \frac{\partial}{\partial t}(\sum_{i=1}^{N_p}\hat u_h \phi_i) \phi_j+ \frac{\partial \phi_j}{\partial x} (\sum_{i=1}^{N_p}\hat u_h \phi_i) = -[(au_h)\phi_j]^{k+1}_{k}$$
where in this last step, I'm confused about two things
- should I replace $u_h$ in the RHS?
- From there how can a matrix arise?
I'm sure I am missing something in my development but I cannot see what it is.