VEM Error Equation

math

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We start from the Green's formula

$$-(\Delta u, v)_K = (\nabla u, \nabla v)_K + \langle \nabla u\cdot n, v \rangle_{\partial K}.$$
We will apply it to $u,v,\Pi u, Q_Kv$ and use the definition of $H^1$ projection $\Pi_K$ and $L^2$ projection $Q_K$ to get
$$\begin{align*} -(\Delta u, Q_K v)_K &= (\nabla \Pi u, \nabla Q_K v)_K + \langle \nabla u\cdot n, Q_K v \rangle_{\partial K}\\ (\Delta \Pi u, Q_K v)_K &= -(\nabla \Pi u, \nabla Q_K v)_K - \langle \nabla \Pi u\cdot n, Q_K v \rangle_{\partial K}\\ - (\Delta \Pi u, v)_K &= (\nabla \Pi u, \nabla v)_K + \langle \nabla \Pi u\cdot n, Q_F v \rangle_{\partial K} \end{align*}$$
Adding together and notice that $(\Delta \Pi u, Q_K v)_K = (\Delta \Pi u, v)_K$ by definition of $Q_K$, we get
$$\begin{align*} -(\Delta u, Q_K v)_K = &(\nabla \Pi u, \nabla v)_K \\ + & \langle (\nabla u -\nabla \Pi u)\cdot n, Q_K v \rangle_{\partial K} \\ + & \langle \nabla \Pi u\cdot n, Q_F v \rangle_{\partial K}. \end{align*}$$
By definition, the first term can be written as $(\nabla \Pi u, \nabla \Pi v)_K$. By the continunity of $\nabla u\cdot n$ across the sides of elements, and $Q_Fv$ is single value at a face, $\sum_{K} \langle \nabla u \cdot n, Q_F v \rangle_{\partial K} = 0$. Therefore we arrive at the following integration by part formula
$$\begin{align*} -\sum_{K}(\Delta u, Q_K v)_K = & \sum_{K}(\nabla \Pi u, \nabla \Pi v)_K \\ + & \sum_{K} \langle (\nabla u -\nabla \Pi u)\cdot n, Q_K v - Q_F v \rangle_{\partial K}. \end{align*}$$