Unfortunately, I could not achieve desired orders of accuracy but results are converged. My guess is that the way to measure L1 norm is not right. Could anyone advice me how to measure L1 norm in DGM applications?

Hi, Jinwon,

I'm also new to the world of DGFEM, so don't take my words for granted. L1, L2, L_inf, etc. norms in DGFEM are defined in pretty the same way they're defined in FEM. Maybe your problem could be that in DGFEM the solution in each element is an high order polynomial: if you limit yourself to second order, then there's no reason to go DGFEM because the number of degrees of freedom is way higher than for FEM, and the nonlinear stability could be achieved more cheaply using streamline upwinding or other methods. Due to the high order of the polynomial approximation, you can't compute the L1 norm using simply the value of the solution at the element centroid times the element measure. You have to compute the integral of the absolute valute of the solution over each element and sum all the integrals. Depending on the problem that you are solving, there may be ways to reduce the computational cost of the integration (expecially true for linear problems). However, I would first of all look at the way the numerical flux, the boundary conditions, the numerical integration and the timestepping are implemented in your code. DG isn't trivial to code and there's a lot which needs to be taken care of in order to get the correct high convergence rates. However it is necessary to do that, because using DG without getting the high accuracy is basically pointless.

Best regards,

Sergio