# Discontinuous Galerkin for transport equation with non-constant advection

This question is mainly an inquiry about the usefulness of Discontinuous Galerkin (DG) for the time-independent transport equation of the form $$\sigma u+\beta\cdot\nabla u =f,\;\;\;\text{on }\Omega\subset\mathbb{R}^2$$ when the advection field $$\beta$$ is a non-constant function of $$x$$ (or $$u$$). Boundary conditions are $$u=g\;\;\;\text{ on the 'inflow' boundary }\;\;\Gamma_-:=\{x\in\partial\Omega:\beta(x)\cdot\mathbf{n}(x)<0\}.$$ Now let $$\mathcal{T}_h$$ be a triangulation of $$\Omega$$ and $$V_h=\{v\in L^2(\Omega):v_{|K}\in\mathbb{P}_d(K),\forall K\in\mathcal{T}_h\}$$ be the usual DG space. Then the local formulation on each triangle $$K\in\mathcal{T}_h$$ is given by $$\sigma\int_K u_hv +\int_K (\beta\cdot\nabla u_h)v-\int_K fv=\int_{\partial K^-} (\beta\cdot\mathbf{n}_K)(u_h-\xi_h)v,$$ where $$\partial K^-$$ is in the inflow boundary and $$\xi_h=\begin{cases}0 &\text{ on }\;\;\partial K^-\cap \Gamma_-,\\ \beta\cdot{n}_K(x)\lim_{\epsilon\to 0^+}u_h(x-\epsilon\beta)&\text{ on }\;\;\partial K^-\setminus\Gamma_-. \end{cases}$$

In the case of when $$\beta$$ is constant, Lesaint and Raviart [1] showed that there is an enumeration of the triangles following the characteristic direction $$\beta$$ which renders the matrix of the discrete problem block triangular. I understand the interest of DG over CG (Continuous Galerkin) here, where it is easier to invert the system matrix, thus more or less justifying the extra d.o.f (degrees of freedom).

However, I don't think this advantage remains when $$\beta$$ becomes non-constant and has loops (if $$\beta$$ is an electric field for instance). Finding a proper enumeration becomes harder and there is more global coupling. So why should I use DG with its extra d.o.f? Am I missing some fix that makes DG advantageous again?

• Isn't $\xi_h$ equal to $g$ on the boundary (instead of 0)? Commented Nov 23, 2023 at 9:40
• @KennethAssogba You are right, It is a mistake on my part. It should've been $u=0$ on $\Gamma_-$ Commented Nov 26, 2023 at 13:53

## 1 Answer

If the exact solution of the transport problem is not smooth, the numerical solutions using the continuous finite element method may exhibit spurious oscillations around the discontinuity. As you said, with the discontinuous Galerkin method, the continuity constraint of the $$V_h$$ space is removed. The functions of $$V_h$$ can be discontinuous at the interfaces between two cells.

You will find below an image from Claes Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method (1987). I recommend chapter 9.

• By the way, in your question, isn't $\xi$ equal to $g$ on the border (instead of 0)? Commented Nov 22, 2023 at 16:04
• Something is weird in the graph on the left. I know usually CG methods like standard FEM are only piecewise continuous but the CG approximation looks like a smooth Fourier approximation. Thus, it is quite expected to oscillate when approximating a step function. However, I want a comparison with the standard continuous FEM. Commented Nov 26, 2023 at 14:00
• In addition, even if you use standard continuous FEM with the same mesh size, I believe the graph would look a lot like the one right. So my question is still not answered which is why then choose DG over the standard CG-FEM? Commented Nov 26, 2023 at 14:02