1}
				\end{alignat}
				\label{eq:f_bounds_time}
			\end{subequations}
		\end{lemma}
		
		\begin{proof}
		
		
		
			
			
			
			
			
			
		
		
		
			
			
			
			
			
			
			
			The results can be obtained using the Lipschitz continuity of all the coefficients and the smoothness of $p_\ell, \, s_a, \, s_v$.
		\end{proof}
		\begin{lemma}
			For any $0 \leq n \leq N$, and any $w_h \in X_h$, we have the following bounds
			\begin{subequations}
				\begin{alignat}{2}
					\abs{\tilde{b}^{n}_{p}(p_\ell^{n+1}, w_h) - b^{n}_{p}(p_\ell^{n+1}, w_h)} &\leq C \left(h^2 + \Vert S_{a_h}^{n} - s_a^{n} \Vert + 
					\Vert S_{v_h}^{n} - s_v^{n}\Vert\right) \vertiii{w_h},
					\label{eq:bp_bp_tilde}
					\\
					\abs{\tilde{b}^{n}_{a}(s_a^{n+1}, w_h) - b^{n}_{a}(s_a^{n+1}, w_h)} &\leq C \left(h^2 + \Vert S_{a_h}^{n} - s_a^{n} \Vert + \Vert S_{v_h}^{n} - s_v^{n} \Vert\right) \vertiii{w_h},
					\label{eq:ba_ba_tilde}
					\\
					\abs{\tilde{b}^{n}_{v}(s_v^{n+1}, w_h) - b^{n}_{v}(s_v^{n+1}, w_h)} &\leq C \left(h^2 + \Vert S_{a_h}^{n} - s_a^{n} \Vert +
					\Vert S_{v_h}^{n} - s_v^{n}\Vert\right) \vertiii{w_h}.
					\label{eq:bv_bv_tilde}
				\end{alignat}
				\label{eq:b_bounds}
			\end{subequations}
			Moreover,
			\begin{subequations}
				\begin{alignat}{2}
					\abs{\tilde{f}^{n}_{p}(w_h) - f^{n}_{p}(w_h)} &\leq  C \left(h^2 + \Vert S_{a_h}^{n} - s_a^{n} \Vert + 
					\Vert S_{v_h}^{n} - s_v^{n}\Vert\right) \vertiii{w_h},
					\label{eq:fp_fp_tilde}
					\\
					\abs{\tilde{f}^{n}_{a}(w_h) - f^{n}_{a}(w_h)} &\leq C \left(h + \vertiii{e_{p_h}^{n+1}} + \Vert S_{a_h}^{n} - s_a^{n} \Vert + \Vert S_{v_h}^{n} - s_v^{n} \Vert\right) \vertiii{w_h},
					\label{eq:fa_fa_tilde}
					\\
					\abs{\tilde{f}^{n}_{v}(w_h) - f^{n}_{v}(w_h)} &\leq C \left(h + \vertiii{e_{p_h}^{n+1}} + \Vert S_{a_h}^{n} - s_a^{n} \Vert + \Vert S_{v_h}^{n} - s_v^{n} \Vert\right) \vertiii{w_h}.
					\label{eq:fv_fv_tilde}
				\end{alignat}
				\label{eq:f_bounds}
			\end{subequations}
		\end{lemma}
		
		\begin{proof}
			We start by showing \cref{eq:bp_bp_tilde}. Note that using the definition of $b_{p}$ \cref{eq:ah_pressure}, we obtain
		
			\begin{multline}
				\abs{\tilde{b}^{n}_{p}(p_\ell^{n+1}, w_h) - b^{n}_{p}(p_\ell^{n+1}, w_h)}
				\leq \sum_{K \in \mathcal{E}_h}\int_K \abs{\del[1]{\tilde{\lambda}_t^{n} - \lambda_t^{n}} \kappa \nabla p_\ell^{n+1}\cdot \nabla w_h}
				\\
				+ \sum_{e \in \Gamma_h }\int_e \abs{\cbr[1]{\left(\tilde{\lambda}_t^{n} - \lambda_t^{n}\right) \kappa\nabla p_\ell^{n+1}\cdot \boldsymbol{n}_e}\sbr[0]{w_h}}.
				\label{eq:716step}
			\end{multline}
			With the Lipschitz continuity assumptions \ref{hyp_lipschitz}, we can write
			\begin{equation*}
				\abs{\tilde{\lambda}_t^{n} - \lambda_t^{n}}
				\leq C (\vert S_{v_h}^{n} - s_v^{n} \vert + \vert S_{a_h}^{n} - s_a^{n} \vert).
			\end{equation*}
			The first term in the right-hand side of \eqref{eq:716step} is bounded by
			\begin{align*}
				\sum_{K \in \mathcal{E}_h}\int_K \abs{\left(\tilde{\lambda}_t^{n} - \lambda_t^{n}\right) \kappa \nabla p_\ell^{n+1}\cdot \nabla w_h}
				&\leq C \left(\Vert S_{a_h}^{n} - s_a^{n} \Vert + 
				\Vert S_{v_h}^{n} - s_v^{n}\Vert\right) \vertiii{w_h},
			\end{align*}
		
			where we have used the Cauchy--Schwarz inequality, boundedness of $\kappa$ and the smoothness of $p_\ell$. For the second term in the right-hand side of \eqref{eq:716step}, we have
		
			\begin{multline*}
				\sum_{e \in \Gamma_h }\int_e \abs{\cbr[1]{\left(\tilde{\lambda}_t^{n} - \lambda_t^{n}\right) \kappa\nabla p_\ell^{n+1}\cdot \boldsymbol{n}_e}\sbr[1]{w_h}} \leq C \sum_{e \in \Gamma_h }\int_e \cbr[1]{\vert S_{v_h}^{n} - s_v^{n} \vert + \vert S_{a_h}^{n} - s_a^{n} \vert}\abs{\sbr[0]{w_h}} 
				\\
				\leq C \sum_{e \in \Gamma_h }\int_e \cbr[1]{\vert e_{v_h}^n \vert + \vert e_{a_h}^n \vert}\abs{\sbr[0]{w_h}} + C \sum_{e \in \Gamma_h }\int_e \cbr[1]{\vert e_{v_\pi}^n \vert + \vert e_{a_\pi}^n \vert}\abs{\sbr[0]{w_h}}
				\\
				\leq C \sum_{e \in \Gamma_h }h_e^{1/2}\del[1]{\Vert e_{v_h}^n \Vert_{L^2(e)} + \Vert e_{a_h}^n \Vert_{L^2(e)}}h_e^{-1/2}\norm[0]{\sbr[0]{w_h}}_{L^2(e)} + C \sum_{e \in \Gamma_h } h_e^{1/2}\del[1]{\Vert e_{v_\pi}^n \Vert_{L^2(e)} + \Vert e_{a_\pi}^n \Vert_{L^2(e)}}h_e^{-1/2}\norm[0]{\sbr[0]{w_h}}_{L^2(e)}.
			\end{multline*}
			Using the trace inequality \cref{eq:trace_ineq} and the projection estimates \cref{eq:L2projerrors_face}, we have
			\begin{equation*}
				\sum_{e \in \Gamma_h }\int_e \abs{\cbr[1]{\left(\