discretizing surface integral using nodal DG method

Question

I am currently learning nodal DG methods, primarily through the book by Warburton, and am a bit confused on how to handle surface integrals using straight edged elements.  On page 187 (and on page 214) of Warburton's book, we have
$$
int_{partial E_{k}} mathbf{n} cdot f_{h} ell_{i}^{k}(mathbf{x}) dmathbf{x}.
$$
Recall that $f_{h} = sum_{j=1}^{N} f_{j}(x,t)ell_{j}(mathbf{x})$ and $ell_{i}(mathbf{x})$ is our basis function, and so,
$$
int_{partial E_{k}} mathbf{n} cdot f_{h} ell_{i}^{k}(mathbf{x}) dmathbf{x} = int_{partial E_{k}} mathbf{n} cdot left[ sum_{j=1}^{N} f_{j}(x,t)ell_{j}(mathbf{x}) right]ell_{i}^{k}(mathbf{x}) dmathbf{x}
= sum_{j=1}^{N} mathbf{n} cdot f_{j}(x,t) int_{partial E_{k}} ell{j}(mathbf{x}) ell_{i}(mathbf{x}) d mathbf{x}
= sum_{j=1}^{N} mathbf{n} cdot f_{j}(x,t) M^{k}_{ij}.
$$
Now, to discretize the surface mass matrix, we must use a quadrature rule on each face (since I am working in 2D it's a line integral along each edge).  With that said, since we have an orthonormal basis, we can compute the mass matrix analytically.  For volume integrals, Warburton shows that $M^{k}_{ij} =  (mathcal{V} mathcal{V}^{T})^{-1} J,$ where $J$ is the metric Jacobian.  So intuitively, the same would be for the mass matrix, however, the Vandermonde matrix, $mathcal{V}$ would be constructed so its on the local face points $(r,s)$, where $(r,s) in [-1,1]$. Now, looking at their code (https://github.com/tcew/nodal-dg/blob/master/Codes1.1/CFD2D/CurvedEulerRHS2D.m), we see that the surface integral term has no surface mass matrix.  There is an inverse mass matrix applied to the numerical flux but it is the one from the time-derivative.  I was curious if someone could point out why there is no surface mass matrix in their formulation.

ConvexHull · Accepted Answer

In 2D your volume part consists of a double integral, whereas the surface part is a standard line integral. Generally this is described with tensorial notations.
If you consider the Cartesian case, the volume part results in a mass matrix similar to
$M_{Omega}equiv  mathbf{M}_{text{1}} otimes mathbf{M}_{text{1}}$,
whereas the surface part results in a mass matrix similar to
$M_{partial Omega}^{xi}  equiv mathbf{M}_{text{1}} otimes mathbf{I}_1
 M_{partial Omega}^{eta} equiv mathbf{I}_1 otimes mathbf{M}_{text{1}}$.
Here $mathbf{I}_1$ is the identity matrix.
If you invert the volume part and apply it on $xi$-faces you will get something like
$=left( mathbf{M}_{text{1}} otimes mathbf{M}_{text{1}}right)^{-1}(mathbf{M}_{text{1}} otimes mathbf{I}_1)  =  (mathbf{M}_{text{1}} ^{-1} otimes mathbf{M}_{text{1}} ^{-1})   (mathbf{M}_{text{1}} otimes mathbf{I}_1)  = (mathbf{M}_{text{1}} ^{-1}  mathbf{M}_{text{1}} ) otimes   (mathbf{M}_{text{1}}^{-1} mathbf{I}_1) = mathbf{I}_{text{1}} otimes   mathbf{M}_{text{1}}^{-1}$
or similar on $eta$-faces
$=left( mathbf{M}_{text{1}} otimes mathbf{M}_{text{1}}right)^{-1}(mathbf{I}_{text{1}} otimes mathbf{M}_1)  =  (mathbf{M}_{text{1}} ^{-1} otimes mathbf{M}_{text{1}} ^{-1})   (mathbf{I}_{text{1}} otimes mathbf{M}_1)  = (mathbf{M}_{text{1}} ^{-1}  mathbf{I}_{text{1}} ) otimes   (mathbf{M}_{text{1}}^{-1} mathbf{M}_1) = mathbf{M}_{text{1}}^{-1} otimes   mathbf{I}_{text{1}}.$
Summerizing: I think you are missing the fact, that both mass matrices (volume and surface) cancel, resulting in a 1D inverse mass matrix. Note that numerical fluxes on $eta$-faces are projected into the volume in $xi$-direction, whereas fluxes on $xi$-faces are projected into the volume in $eta$-direction.
Regards

discretizing surface integral using nodal DG method

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