0 & = (D_{\Delta t}^\alpha (M^{n+1}  -\tilde{M}^{n+1}),W)_2 - ((M^{n+1}-\tilde{M}^{n+1}), \Delta_h W)_2 \\
& + \sum_{i,j} M_{i,j}^{n+1} [D_hW]_{i,j} \cdot \nabla_q g ( x_{i,j},[D_hU^{n}])  - \sum_{i,j} \tilde{M}_{i,j}^{n+1} [D_hW]_{i,j} \cdot \nabla_q g  (x_{i,j},[D_h\tilde{U}^{n}] ).
\end{aligned}$$

Let $Z = M^{n+1} - \tilde{M}^{n+1}$ in and $W = U^{n} - \tilde{U}^{n}$ in . Subtracting the two equations so obtained, summing over $n=0,1,\dots,N-1$ and recalling that, by Lemma \[le:by\_parts\_n\_n\_plus\_1\], we have $$\sum_{n=0}^{N-1} (\bar D_{\Delta t}^{\alpha} (U^{n} -\tilde{U}^{n}),M^{n+1} - \tilde{M}^{n+1})_2 =  \sum_{n=0}^{N-1} (D_{\Delta t}^\alpha (M^{n+1}  -\tilde{M}^{n+1}),U^{n} - \tilde{U}^{n})_2,$$ we finally get $$\label{eq:un_nonneg_terms}
\begin{aligned}
0 & = \sum_{n=0}^{N-1}   \bigg(f_h[M^{n+1}] - f_h[\tilde{M}^{n+1}],M^{n+1} - \tilde{M}^{n+1}\bigg)_2 \\
& +   \sum_{n=0}^{N-1}  \sum_{i,j} M_{i,j}^{n+1}\bigg( g(x_{i,j},[D_h\tilde{U}^{n}]_{i,j}) - g(x_{i,j},[D_hU^{n}]_{i,j})  \\
& -[D_h(\tilde{U}^{n}-U^{n})]_{i,j} \cdot \nabla_q g ( x_{i,j},[D_hU^{n}])   \bigg)\\
& + \sum_{n=0}^{N-1}  \sum_{i,j} \tilde{M}_{i,j}^{n+1} \bigg(g(x_{i,j},[D_hU^{n}]_{i,j}) - g(x_{i,j},[D_h\tilde{U}^{n}]_{i,j}) \\
& -[D_h(U^{n}-\tilde{U}^{n})]_{i,j} \cdot \nabla_q g ( x_{i,j},[D_h\tilde{U}^{n}] )  \bigg).
\end{aligned}$$ By the monotonicity of $f_h[\cdot]$ and assumption (G4), all three terms on RHS of are non negative. Hence, we obtain that $M^{n} = \tilde M^{n}$ for all $0 \le n \le N$. By Prop. \[prop:ex\_HJ\], it also follows that $U^{n} = \tilde U^{n}$ for all $0 \le n \le N$.

Existence
---------

To show the existence of a solution for the scheme , we rely on a fixed point 