1. **Problem Statement:** Verify if the given proof establishes the stability of the Interior Penalty Discontinuous Galerkin (IPDG) bilinear form $a(\cdot,\cdot)$ defined on a discontinuous finite element space $V_h$ with penalty parameter $\sigma>0$.
2. **Recall the bilinear form:**
$$
a(u,v) = \sum_{K\in\mathcal{T}_h} \int_K \nabla u \cdot \nabla v \, dx - \sum_{e\in\mathcal{E}_h} \int_e \{\nabla u\} \cdot n_e [v] \, ds + \theta \sum_{e\in\mathcal{E}_h} \int_e \{\nabla v\} \cdot n_e [u] \, ds + \sum_{e\in\mathcal{E}_h} \int_e \frac{\sigma}{h_e} [u][v] \, ds.
$$
where $\theta \in \{-1,0,1\}$ corresponds to SIPG, IIPG, and NIPG methods.
3. **Goal:** Show there exists $C>0$ independent of mesh size $h$ such that
$$
a(v,v) \ge C \|v\|_{DG}^2 \quad \forall v \in V_h,
$$
where
$$
\|v\|_{DG}^2 = \sum_{K\in\mathcal{T}_h} \|\nabla v\|_{L^2(K)}^2 + \sum_{e\in\mathcal{E}_h} \frac{1}{h_e} \|[v]\|_{L^2(e)}^2.
$$
4. **Step-by-step verification:**
- Substitute $u=v$ in $a(u,v)$:
$$
a(v,v) = \sum_{K\in\mathcal{T}_h} \|\nabla v\|_{L^2(K)}^2 + (\theta - 1) \sum_{e\in\mathcal{E}_h} \int_e \{\nabla v\} \cdot n_e [v] \, ds + \sum_{e\in\mathcal{E}_h} \int_e \frac{\sigma}{h_e} [v]^2 \, ds.
$$
- Use Cauchy--Schwarz and trace inequalities to bound the flux term:
$$
\left| \int_e \{\nabla v\} \cdot n_e [v] \, ds \right| \le \varepsilon \|\nabla v\|_{L^2(K_e)}^2 + C_\varepsilon \frac{1}{h_e} \|[v]\|_{L^2(e)}^2
$$
for any $\varepsilon > 0$.
- **Case 1: SIPG ($\theta = -1$)**
- Then $(\theta - 1) = -2$.
- The negative flux term can be controlled by choosing $\varepsilon$ small and $\sigma$ large.
- This yields coercivity:
$$
a(v,v) \ge C \|v\|_{DG}^2.
$$
- **Case 2: IIPG ($\theta = 0$)**
- Then $(\theta - 1) = -1$.
- Similar argument as SIPG applies.
- For sufficiently large $\sigma$, penalty dominates, ensuring coercivity.
- **Case 3: NIPG ($\theta = 1$)**
- Then $(\theta - 1) = 0$.
- Flux term vanishes.
- Coercivity follows immediately:
$$
a(v,v) = \sum_{K\in\mathcal{T}_h} \|\nabla v\|_{L^2(K)}^2 + \sum_{e\in\mathcal{E}_h} \int_e \frac{\sigma}{h_e} [v]^2 \, ds \ge C \|v\|_{DG}^2.
$$
5. **Conclusion:** The proof correctly applies standard inequalities and parameter choices to establish the stability (coercivity) of the IPDG bilinear form for all three cases of $\theta$. The argument is rigorous and standard in DG theory.
**Final answer:** Yes, the proof is correct and valid for the stability of the DG scheme.
Ipdg Stability 59Fd60
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