 defined above is a very general mesh/partition on $\Omega$, as we do not specify the shape and conformal property of $K\in \mathcal{T}_h$. The interior penalty discontinuous Galerkin (IPDG) method can be extended to such a general mesh, without any modification of the formulation. However, to ensure its approximation rate, certain conditions must be imposed on $\mathcal{T}_h$ and the discrete function spaces. In this paper, we are interested in discussing the minimum requirements of such conditions. First, we shall give the formulation of the interior penalty discontinuous Galerkin method.

Let $V_K$ be a finite dimensional space of smooth functions defined on $K \in \mathcal{T}_h$. Define $$V_h=\{v\in L^2(\Omega):v|_K\in V_K,\textrm{ for all } K\in\mathcal{T}_h\},$$ and $$V(h)=V_h+\left( H_0^1(\Omega)\cap \prod_{K\in\mathcal{T}_h} H^{r}(K) \right),\qquad \textrm{where }r>\frac{3}{2}.$$ For any internal interface $e = \bar{K_1}\cap\bar{K_2} \in \mathcal{E}_h$, let $\bn_1$ and $\bn_2$ be the unit outward normal vectors on $e$, associated with ${K_1}$ and ${K_2}$, respectively. For $v\in V(h)$, define the average $\{\nabla v\}$ and jump $[v]$ on $e$ by $$\{\nabla v\} = \frac{1}{2}\left(\nabla v|_{K_1} + \nabla v|_{K_2} \right),\qquad
[v] = v|_{K_1} \bn_1 + v|_{K_2} \bn_2.$$ On any boundary segment $e= \bar{K}\cap\partial\Omega$, the above definitions of average and jump need to be modified: $$\{\nabla v\} = \nabla v|_{K},\qquad  [v] = v|_{K} \bn_K,$$ where $\bn_K$ is the unit outward normal vector on $e$ with respect to $K$.

Define a bilinear form on $V(h)\times V(h)$ by $$\begin{aligned}
  A(u,v) =& \sum_{K\in \mathcal{T}_h}(\nabla u,\nabla v)_K-\sum_{e\in\mathcal{E}_h}\langle\{\nabla u\},\, [v]\rangle_e \\
  &\quad -\delta \sum_{e\in\mathcal{E}_h}\langle\{\nabla v\},\, [u]\rangle_e + \alpha \sum_{e\in\mathcal{E}_h}\frac{1}{h_e} \langle[u],\, [v]\rangle_e,
\end{aligned}$$ where $\delta = \pm 1,\, 0$ and $\alpha>0$. when $\delta=1$, the bilinear form $A(\cdot,\cdot)$ is symmetric. The constant $\alpha$ is usually