Let the time interval \([0,T]\) be divided into \(\ell _{\kappa }\) intervals of length κ, where κ tends to zero as \(\ell _{\kappa }\to \infty \). Let \({\mathbf{1}}_{I_{i}}\) be the characteristic function of \(I_{i}=[i\kappa,(i+1)\kappa )\), \(i=0,\dots,\ell _{\kappa }\). We define a set of piecewise-constant in time functions by
$$\begin{aligned} \mathbb{L}_{\kappa }= \Biggl\{ w_{\kappa }(\pmb{x},t)= \sum_{i=1}^{\ell _{\kappa }}{\mathbf{1}}_{I_{i}}(t) \varphi _{i}(\pmb{x}): \varphi \in C_{0}^{2}( \overline{\Omega }) \text{ and $\varphi \leq \chi $ in $\Omega $ a.e.} \Biggr\} . \end{aligned}$$
(4.1)
Lemma 4.1
For \(T>0\), let \(({\mathcal{D}})_{m\in {\mathbb{N}}}\) be a sequence of gradient discretizations that is consistent. Let \(\bar{w}_{\kappa }\in \mathbb{L}_{\kappa }\) be a piecewise constant in time function, where \(\mathbb{L}_{\kappa }\) is the set defined by (4.1). Then there exists a sequence \((w_{m})_{m\in {\mathbb{N}}}\) such that \(w_{m}=(w_{m}^{(n)})_{n=0,\dots,N_{m}} \in {\mathcal{K}}_{{\mathcal{D}}_{m}}^{N_{m}+1}\) for all \(m\in {\mathbb{N}}\), and, as \(m\to \infty \),
$$\begin{aligned} &\Pi _{{\mathcal{D}}_{m}}w_{m} \to \bar{w}_{\kappa }\quad\textit{strongly in } L^{2}\bigl( \Omega \times (0,T)\bigr), \end{aligned}$$
(4.2a)
$$\begin{aligned} &\nabla _{{\mathcal{D}}_{m}}w_{m} \to \nabla \bar{w}_{\kappa }\quad\textit{strongly in } L^{2}\bigl(\Omega \times (0,T)\bigr)^{d}. \end{aligned}$$
(4.2b)
Proof
Write \(\bar{w}_{\kappa }(\pmb{x},t)=\sum_{i=1}^{\ell _{\kappa }} {\mathbf{1}}_{I_{i}}(t) \phi _{i}(\pmb{x})\), where \(\phi _{i} \in C_{0}^{\infty }(\overline{\Omega })\cap {\mathcal{K}}\). Let \(s\in (0,T)\) and choose \(n:=n(s)\) such that \(s\in (t^{(n(s))},t^{(n(s)+1)}]\). Let \(w_{m}\in X_{{\mathcal{D}}_{m},0}\) be defined by \(w_{m}= \sum_{i=1}^{\ell _{\kappa }}{\mathbf{1}}_{I_{i}}(t^{(n(s)+1)})P_{{ \mathcal{D}}_{m}}\phi _{i}\), where
$$\begin{aligned} P_{{\mathcal{D}}_{m}}(\phi ) ={ \mathop {\operatorname {argmin}}_{\omega \in { \mathcal{K}}_{{\mathcal{D}}_{m}} } \bigl( \Vert \Pi _{{\mathcal{D}}_{m}} \omega - \phi \Vert _{L^{2}(\Omega )} + \Vert \nabla _{{\mathcal{D}}_{m}} \omega - \nabla \phi \Vert _{L^{2}(\Omega )^{d}} \bigr)}. \end{aligned}$$
(4.3)
For \(i=1,\dots,\ell _{\kappa }\), we define \({\xi }_{m}^{i}:(0,T) \to {\mathbb{R}}\) by \({\xi }_{m}^{i}(s)={\mathbf{1}}_{I_{i}}(t^{(n(s)+1)})\) for \(s\in (0,T)\). Using the relation \(ab-cd=(a-c)b+c(b-d)\), we obtain, for all \(s\in (0,T)\) and a.e. \(\pmb{x}\in \Omega \),
$$\begin{aligned} \begin{aligned} (\Pi _{{\mathcal{D}}_{m}}w_{m}- \bar{w}_{\kappa }) (\pmb{x},s)={}& \sum_{i=1}^{\ell _{\kappa }} \bigl({\xi }_{m}^{i}(s)-{\mathbf{1}}_{I_{i}}(s) \bigr)\Pi _{{\mathcal{D}}_{m}}P_{{\mathcal{D}}_{m}}\phi _{i}(\pmb{x}) \\ & {}+\sum_{i=1}^{\ell _{\kappa }} {\mathbf{1}}_{I_{i}}(s) ( \Pi _{{ \mathcal{D}}_{m}}P_{{\mathcal{D}}_{m}}\phi _{i} -\phi _{i} ) ( \pmb{x}). \end{aligned} \end{aligned}$$
An application of the definition of \(S_{{\mathcal{D}}_{m}}\) yields
$$\begin{aligned} \begin{aligned} \Vert \Pi _{{\mathcal{D}}_{m}}w_{m}- \bar{w}_{\kappa } \Vert _{L^{2}( \Omega \times (0,T))}\leq{}& \sum _{i=1}^{\ell _{\kappa }} \bigl\Vert {\xi }_{m}^{i}(s)-{ \mathbf{1}}_{I_{i}}(s) \bigr\Vert _{L^{2}(0,T)} \Vert \Pi _{{\mathcal{D}}_{m}}P_{{ \mathcal{D}}_{m}}\phi _{i} \Vert _{L^{2}(\Omega )} \\ &{} +\sum_{i=1}^{\ell _{\kappa }} \bigl\Vert { \mathbf{1}}_{I_{i}}(s) \bigr\Vert _{L^{2}(0,T)} \Vert \Pi _{{\mathcal{D}}_{m}}P_{{\mathcal{D}}_{m}}\phi _{i} -\phi _{i} \Vert _{L^{2}( \Omega )} \\ \leq{}& \sum_{i=1}^{\ell _{\kappa }} \bigl\Vert {\xi }_{m}^{i}(s)-{\mathbf{1}}_{I_{i}}(s) \bigr\Vert _{L^{2}(0,T)} \bigl( S_{{\mathcal{D}}_{m}}(\phi _{i})+ \Vert \phi _{i} \Vert _{L^{2}( \Omega )} \bigr) \\ &{} +C_{1}\sum_{i=1}^{\ell _{\kappa }}S_{{\mathcal{D}}_{m}}( \phi _{i}), \end{aligned} \end{aligned}$$
(4.4)
where \(C_{1}=\sum_{i=1}^{\ell _{\kappa }} \|{\mathbf{1}}_{I_{i}} \|_{L^{2}(0,T)}\). Using consistency, one obtains \(S_{{\mathcal{D}}_{m}}(\phi _{i})\to 0\) as \(m\to \infty \), for any \(i=0,\dots,\ell _{\kappa }\), which implies that the second term on the right-hand side vanishes. In the case in which both s, \(t^{(n(s)+1)} \in I_{i}\) or both s, \(t^{(n(s)+1)} \notin I_{i}\), the quantity \({\xi }_{m}^{i}(s)-{\mathbf{1}}_{I_{i}}(s)\) equals zero. In the case in which \(s\in I_{i}\) and \(t^{(n(s)+1)} \notin I_{i}\) or \(s\notin I_{i}\) and \(t^{(n(s)+1)} \in I_{i}\), one can deduce, writing \(I_{i}=[a_{i},b_{i}]\) and because s is chosen such that \(|s-t^{(n(s)+1)}|\leq \delta t_{{\mathcal{D}}_{m}}\),
$$\begin{aligned} \bigl\Vert {\xi }_{m}^{i}(s)-{\mathbf{1}}_{I_{i}}(s) \bigr\Vert _{L^{2}(0,T)}^{p} &\leq { \mathrm{measure}} \bigl([a_{i}-\delta t_{{\mathcal{D}}_{m}},a_{i}+\delta t_{{ \mathcal{D}}_{m}}] \cup [b_{i}-\delta t_{{\mathcal{D}}_{m}}, b_{i}+ \delta t_{{\mathcal{D}}_{m}}]\bigr) \\ &\leq 4 \delta t_{{\mathcal{D}}_{m}}. \end{aligned}$$
This shows that the first term on the right-hand side of (4.4) tends to zero when \(m \to \infty \). Hence, (4.2a) is concluded. The proof of (4.2b) is obtained by the same reasoning, replacing \(\bar{w}_{\kappa }\) by \(\nabla \bar{w}_{\kappa }\) and \(\Pi _{{\mathcal{D}}_{m}}w_{m}\) by \(\nabla _{{\mathcal{D}}_{m}}w_{m}\). □
Lemma 4.2
(Energy estimates)
Let Hypothesis 2.1hold. If \({\mathcal{D}}\) is a gradient discretization such that \(\delta _{\mathcal{D}}< \frac{1}{2M}\), \({\mathcal{K}}_{\mathcal{D}}\) is a nonempty set, and \((A,B)\in {\mathcal{K}}_{\mathcal{D}}\times X_{{\mathcal{D}},0}\) is a solution of the approximate scheme (3.4a)–(3.4b), then there exists a constant \(C_{2}\geq 0\) only depending on Ω, \(d_{1}\), T, M, \(C_{0}:=\max (F({\mathbf{0}}), G({\mathbf{0}}))\), \(\| \Pi _{{\mathcal{D}}}I_{\mathcal{D}}A_{\mathop{\operatorname{ini}}} \|_{L^{2}(\Omega )}\), \(\| \nabla _{{\mathcal{D}}}I_{\mathcal{D}}\nabla A_{\mathop{ \operatorname{ini}}} \|_{L^{2}(\Omega )^{d}}\), and \(\| \Pi _{{\mathcal{D}}}J_{\mathcal{D}}B_{\mathop{\operatorname{ini}}} \|_{L^{2}(\Omega )}\), such that
$$\begin{aligned} \begin{aligned} &\Vert \delta _{\mathcal{D}}A \Vert _{L^{2}(\Omega \times (0,T))} + \Vert \nabla _{ \mathcal{D}}A \Vert _{L^{\infty }(0,T;L^{2}(\Omega )^{d})} \\ &\quad{}+ \Vert \Pi _{\mathcal{D}}B \Vert _{L^{\infty }(0,T;L^{2}(\Omega ))}+ \Vert \nabla _{ \mathcal{D}}B \Vert _{L^{2}(\Omega \times (0,T))^{d}}\leq C_{2}. \end{aligned} \end{aligned}$$
(4.5)
Proof
We start by taking \(\varphi:=A^{(n)}\) (it belongs to \({\mathcal{K}}_{\mathcal{D}}\)) and the function \(\psi:=\delta t^{ (n+\frac{1}{2}) }B^{(n+1)}\) in (3.4a)–(3.4b) to get
$$\begin{aligned} \begin{aligned} &\delta t^{(n+\frac{1}{2})} \int _{\Omega } \bigl\vert \delta _{\mathcal{D}}^{(n+ \frac{1}{2})}A \bigr\vert ^{2} \,\mathrm{d}\pmb{x}+ \int _{\Omega }{\mathbf {D}_{A}} \nabla _{\mathcal{D}}A^{(n+1)} \cdot \nabla _{\mathcal{D}}\bigl(A^{(n+1)}-A^{(n)}\bigr) \,\mathrm{d}\pmb{x} \\ &\quad\leq \delta t^{(n+\frac{1}{2})} \int _{\Omega }F\bigl(\Pi _{\mathcal{D}}A^{(n+1)}, \Pi _{\mathcal{D}}B^{(n+1)}\bigr)\delta _{\mathcal{D}}^{(n+\frac{1}{2})}A \,\mathrm{d}\pmb{x}, \end{aligned} \end{aligned}$$
(4.6)
and
$$\begin{aligned} \begin{aligned} & \int _{\Omega } \bigl(\Pi _{\mathcal{D}}B^{(n+1)}(\pmb{x})- \Pi _{ \mathcal{D}}B^{(n)}(\pmb{x}) \bigr) \Pi _{\mathcal{D}}B^{(n+1)}( \pmb{x}) \,\mathrm{d}\pmb{x} \\ &\qquad{}+ \int _{t^{(n)}}^{t^{(n+1)}} \int _{\Omega }{\mathbf {D}_{B}} \bigl\vert \nabla _{ \mathcal{D}}B^{(n+1)}(\pmb{x}) \bigr\vert ^{2} \,\mathrm{d} \pmb{x} \,\mathrm{d}t \\ &\quad= \delta t^{(n+\frac{1}{2})} \int _{\Omega }G\bigl(\Pi _{\mathcal{D}}A^{(n+1)}( \pmb{x}), B^{(n+1)}(\pmb{x})\bigr) \Pi _{\mathcal{D}}B^{(n+1)}(\pmb{x}) \,\mathrm{d}\pmb{x}. \end{aligned} \end{aligned}$$
(4.7)
Applying the fact that \((r-s)\cdot r \geq \frac{1}{2}|r|^{2}-\frac{1}{2}|s|^{2}\) to the second term on the left-hand side of (4.6) and to the first term on the left-hand side of (4.7), it follows that
$$\begin{aligned} \begin{aligned} &\delta t^{(n+\frac{1}{2})} \int _{\Omega } \bigl\vert \delta _{\mathcal{D}}^{(n+ \frac{1}{2})}A \bigr\vert ^{2} \,\mathrm{d}\pmb{x}+\frac{d_{1}}{2} \int _{\Omega } \bigl( \bigl\vert \nabla _{\mathcal{D}}A^{(n+1)} \bigr\vert ^{2} - \bigl\vert \nabla _{\mathcal{D}}A^{(n)} \bigr\vert ^{2} \bigr) \,\mathrm{d}\pmb{x} \\ &\quad\leq \delta t^{(n+\frac{1}{2})} \int _{\Omega }F\bigl(\Pi _{\mathcal{D}}A^{(n+1)}, \Pi _{\mathcal{D}}B^{(n+1)}\bigr)\delta _{\mathcal{D}}^{(n+\frac{1}{2})}A \,\mathrm{d}\pmb{x}, \end{aligned} \end{aligned}$$
and
$$\begin{aligned} \begin{aligned} &\frac{1}{2} \int _{\Omega } \bigl[ \bigl\vert \Pi _{\mathcal{D}}B^{(n+1)}( \pmb{x}) \bigr\vert ^{2}- \bigl\vert \Pi _{\mathcal{D}}B^{(n)}( \pmb{x}) \bigr\vert ^{2} \bigr] \,\mathrm{d}\pmb{x}+d_{1} \int _{t^{(n)}}^{t^{(n+1)}} \int _{\Omega } \bigl\vert \nabla _{\mathcal{D}}B^{(n+1)}( \pmb{x}) \bigr\vert ^{2} \,\mathrm{d}\pmb{x} \,\mathrm{d}t \\ &\quad\leq \delta t^{(n+\frac{1}{2})} \int _{\Omega }G\bigl(\Pi _{\mathcal{D}}U^{(n+1)}( \pmb{x}), B^{(n+1)}(\pmb{x})\bigr) \Pi _{\mathcal{D}}B^{(n+1)}(\pmb{x}) \,\mathrm{d}\pmb{x}. \end{aligned} \end{aligned}$$
Summing the above inequalities over \(n\in [0,m-1]\), where \(m=0,\dots,N\) gives
$$\begin{aligned} \begin{aligned}& \Vert \delta _{\mathcal{D}}A \Vert _{L^{2}(\Omega \times (0,t^{(m)}))}^{2}+ \frac{d_{1}}{2} \bigl( \bigl\Vert \nabla _{\mathcal{D}}A^{(m)} \bigr\Vert _{L^{2}(\Omega )^{d}}^{2}- \bigl\Vert \nabla _{\mathcal{D}}A^{(0)} \bigr\Vert _{L^{2}(\Omega )^{d}}^{2} \bigr) \\ &\quad\leq \sum_{n=0}^{m-1}\delta t^{(n+\frac{1}{2})} \int _{\Omega }F\bigl(\Pi _{ \mathcal{D}}A^{(n+1)},\Pi _{\mathcal{D}}B^{(n+1)}\bigr) \delta _{\mathcal{D}}^{(n+ \frac{1}{2})} A \,\mathrm{d}\pmb{x}, \end{aligned} \end{aligned}$$
(4.8)
and
$$\begin{aligned} \begin{aligned} &\frac{1}{2} \bigl( \bigl\Vert \Pi _{\mathcal{D}}B^{(m)} \bigr\Vert _{L^{2}(\Omega )}^{2}- \bigl\Vert \Pi _{\mathcal{D}}B^{(0)} \bigr\Vert _{L^{2}(\Omega )}^{2} \bigr) +d_{2}\sum_{n=0}^{m-1} \delta t^{(n+\frac{1}{2})} \bigl\Vert \nabla _{\mathcal{D}}B^{(n)} \bigr\Vert _{L^{2}( \Omega )^{d}}^{2} \\ &\quad\leq \sum_{n=0}^{m-1}\delta t^{(n+\frac{1}{2})} \int _{\Omega }G\bigl(\Pi _{ \mathcal{D}}A^{(n+1)}(\pmb{x}), B^{(n+1)}(\pmb{x})\bigr) \Pi _{\mathcal{D}}B^{(n+1)}( \pmb{x}) \,\mathrm{d}\pmb{x}. \end{aligned} \end{aligned}$$
(4.9)
This, together with the Cauchy–Schwarz inequality, implies that
$$\begin{aligned} \begin{aligned} & \Vert \delta _{\mathcal{D}}A \Vert _{L^{2}(\Omega \times (0,t^{(m)}))}^{2} + \frac{d_{1}}{2} \bigl( \bigl\Vert \nabla _{\mathcal{D}}A^{(m)} \bigr\Vert _{L^{2}(\Omega )^{d}}^{2}- \bigl\Vert \nabla _{\mathcal{D}}A^{(0)} \bigr\Vert _{L^{2}(\Omega )^{d}}^{2} \bigr) \\ &\quad\leq \sum_{n=0}^{m}\delta t^{(n+\frac{1}{2})} \bigl\Vert F\bigl(\Pi _{\mathcal{D}}A^{(n+1)}, \Pi _{\mathcal{D}}B^{(n+1)}\bigr) \bigr\Vert _{L^{2}(\Omega \times (0,T))} \bigl\Vert \delta _{ \mathcal{D}}^{(n+\frac{1}{2})} A \bigr\Vert _{L^{2}(\Omega \times (0,t))} \end{aligned} \end{aligned}$$
and
$$\begin{aligned} \begin{aligned} &\frac{1}{2} \bigl( \bigl\Vert \Pi _{\mathcal{D}}B^{(m)} \bigr\Vert _{L^{2}(\Omega )}^{2}- \bigl\Vert \Pi _{\mathcal{D}}B^{(0)} \bigr\Vert _{L^{2}(\Omega )}^{2} \bigr) +d_{2}\sum_{n=0}^{m-1} \delta t^{(n+\frac{1}{2})} \bigl\Vert \nabla _{\mathcal{D}}B^{(n)} \bigr\Vert _{L^{2}( \Omega )^{d}} \\ &\quad\leq \sum_{n=0}^{m}\delta t^{(n+\frac{1}{2})} \bigl\Vert G\bigl(\Pi _{\mathcal{D}}A^{(n+1)}, \Pi B^{(n+1)}\bigr) \bigr\Vert _{L^{2}(\Omega \times (0,T))} \bigl\Vert \Pi _{\mathcal{D}}B^{(n+1)} \bigr\Vert _{L^{2}(\Omega \times (0,T))}. \end{aligned} \end{aligned}$$
Using the Lipschitz condition, we arrive at
$$\begin{aligned} \begin{aligned} & \Vert \delta _{\mathcal{D}}A \Vert _{L^{2}(\Omega \times (0,t^{(m)}))}^{2} + \frac{d_{1}}{2} \bigl( \bigl\Vert \nabla _{\mathcal{D}}A^{(m)} \bigr\Vert _{L^{2}(\Omega )^{d}}^{2}- \bigl\Vert \nabla _{\mathcal{D}}A^{(0)} \bigr\Vert _{L^{2}(\Omega )^{d}}^{2} \bigr) \\ &\quad\leq \sum_{n=0}^{m}\delta t^{(n+\frac{1}{2})} \bigl\Vert \delta _{\mathcal{D}}^{(n+ \frac{1}{2})} A \bigr\Vert _{L^{2}(\Omega \times (0,t))} \\ &\qquad{} \times \bigl(M \bigl\Vert \Pi _{\mathcal{D}}A^{(n+1)} \bigr\Vert _{L^{2}( \Omega )}+M \bigl\Vert \Pi _{\mathcal{D}}B^{(n+1)} \bigr\Vert _{L^{2}(\Omega )}+ C_{0} \bigr), \end{aligned} \end{aligned}$$
and
$$\begin{aligned} \begin{aligned} &\frac{1}{2} \bigl( \bigl\Vert \Pi _{\mathcal{D}}B^{(m)} \bigr\Vert _{L^{2}(\Omega )}^{2}- \bigl\Vert \Pi _{\mathcal{D}}B^{(0)} \bigr\Vert _{L^{2}(\Omega )}^{2} \bigr) +d_{2}\sum_{n=0}^{m-1} \delta t^{(n+\frac{1}{2})} \bigl\Vert \nabla _{\mathcal{D}}B^{(n)} \bigr\Vert _{L^{2}( \Omega )^{d}} \\ &\quad\leq \sum_{n=0}^{m}\delta t^{(n+\frac{1}{2})} \bigl(M \bigl\Vert \Pi _{ \mathcal{D}}B^{(n+1)} \bigr\Vert _{L^{2}(\Omega )}^{2} \\ &\qquad{} +M \bigl\Vert \Pi _{\mathcal{D}}B^{(n+1)} \bigr\Vert _{L^{2}(\Omega )} \bigl\Vert \Pi _{\mathcal{D}}B^{(n+1)} \bigr\Vert _{L^{2}(\Omega )}+C_{0} \bigl\Vert \Pi _{ \mathcal{D}}B^{(n+1)} \bigr\Vert _{L^{2}(\Omega )} \bigr). \end{aligned} \end{aligned}$$
This, together with Young’s inequality, gives, whenever \(1-\sum_{i=1}^{3}\varepsilon _{i}>0\),
$$\begin{aligned} \begin{aligned} & \Vert \delta _{\mathcal{D}}A \Vert _{L^{2}(\Omega \times (0,t^{(m)}))}^{2} +\frac{d_{1}}{2} \bigl( \bigl\Vert \nabla _{\mathcal{D}}A^{(m)} \bigr\Vert _{L^{2}(\Omega )^{d}}^{2}- \bigl\Vert \nabla _{\mathcal{D}}A^{(0)} \bigr\Vert _{L^{2}(\Omega )^{d}}^{2} \bigr) \\ &\quad\leq \sum_{i=1}^{3}\frac{\varepsilon _{i}}{2} \bigl\Vert \delta _{\mathcal{D}}A^{(n+ \frac{1}{2})} \bigr\Vert _{L^{2}(\Omega \times (0,t))}^{2} \\ &\qquad{} +\sum_{n=0}^{m}\delta t^{(n+\frac{1}{2})} \biggl( \frac{M}{2\varepsilon _{1}} \bigl\Vert \Pi _{\mathcal{D}}A^{(n+1)} \bigr\Vert _{L^{2}( \Omega )}^{2}+\frac{M}{2\varepsilon _{2}} \bigl\Vert \Pi _{\mathcal{D}}B^{(n+1)} \bigr\Vert _{L^{2}(\Omega )}^{2}+ \frac{1}{2\varepsilon _{3}}C_{0} \biggr), \end{aligned} \end{aligned}$$
(4.10)
and
$$\begin{aligned} &\frac{1}{2} \bigl( \bigl\Vert \Pi _{\mathcal{D}}B^{(m)} \bigr\Vert _{L^{2}(\Omega )^{d}}^{2}- \bigl\Vert \Pi _{\mathcal{D}}B^{(0)} \bigr\Vert _{L^{2}(\Omega )^{d}}^{2} \bigr) +d_{2} \sum_{n=0}^{m-1} \delta t^{(n+\frac{1}{2})} \bigl\Vert \nabla _{\mathcal{D}}B^{(n)} \bigr\Vert _{L^{2}(\Omega )^{d}}^{2} \\ &\quad\leq \sum_{n=0}^{m}\delta t^{(n+\frac{1}{2})} \biggl( M\biggl(1+ \frac{1}{2\varepsilon _{4}}\biggr) \bigl\Vert \Pi _{\mathcal{D}}B^{(n+1)} \bigr\Vert _{L^{2}( \Omega )}^{2}\\ &\qquad{} + \frac{\varepsilon _{4}}{2} \bigl\Vert \Pi _{\mathcal{D}}A^{(n+1)} \bigr\Vert _{L^{2}(\Omega )}^{2}+ \frac{\varepsilon _{5}}{2}C_{0}^{2} \biggr). \end{aligned}$$
(4.11)
Thanks to Gronwall inequality [15, Lemma 5.1], inequality (4.11) can be rewritten as
$$\begin{aligned} \begin{aligned} &\frac{1}{2} \bigl\Vert \Pi _{\mathcal{D}}B^{(m)} \bigr\Vert _{L^{2}( \Omega )^{d}}^{2}+d_{2} \sum_{n=0}^{m-1}\delta t^{(n+\frac{1}{2})} \bigl\Vert \nabla _{\mathcal{D}}B^{(n)} \bigr\Vert _{L^{2}(\Omega )^{d}}^{2} \\ &\quad\leq {\mathbf{e}}^{C_{3}} \Biggl(\frac{T\varepsilon _{5}}{2}C_{0}^{2}+ \frac{\varepsilon _{4}}{2}\sum_{n=0}^{m-1}\delta t^{(n+\frac{1}{2})} \bigl\Vert \Pi _{\mathcal{D}}A^{(n+1)} \bigr\Vert _{L^{2}(\Omega \times (0,T))}^{2}+ \frac{1}{2} \bigl\Vert \Pi _{\mathcal{D}}B^{(0)} \bigr\Vert _{L^{2}(\Omega )}^{2} \Biggr), \end{aligned} \end{aligned}$$
where \(C_{3}\) depends on T, M, and \(\varepsilon _{4}\). Combining this inequality with (4.10) yields
$$\begin{aligned} \begin{aligned} &\Biggl(2-\sum_{i=1}^{3} \frac{\varepsilon _{i}}{2} \Biggr) \Vert \delta _{ \mathcal{D}}A \Vert _{L^{2}(\Omega \times (0,t^{(m)}))}^{2} + \biggl( \frac{d_{1}}{2}+\frac{M}{2\varepsilon _{1}}- \frac{\varepsilon _{4}}{2}{\mathbf{e}}^{C_{3}} \biggr) \bigl\Vert \nabla _{\mathcal{D}}A^{(m)} \bigr\Vert _{L^{2}(\Omega )^{d}}^{2} \\ &\qquad{}+\frac{1}{2} \bigl\Vert \Pi _{\mathcal{D}}B^{(m)}(\pmb{x}) \bigr\Vert _{L^{2}(\Omega )}^{2} + (d_{2}-\frac{M}{2\varepsilon _{2}} \bigl\Vert \nabla _{\mathcal{D}}B( \pmb{x},t) \bigr\Vert _{L^{2}(\Omega \times (0,t^{(m)}))^{d}}^{2} \\ &\quad\leq \biggl(\frac{T}{2\varepsilon _{3}}+\frac{T\varepsilon _{5}}{2}{ \mathbf{e}}^{C_{3}} \biggr)C_{0}^{2} +\frac{d_{1}}{2} \bigl\Vert \nabla _{\mathcal{D}}A^{(0)} \bigr\Vert _{L^{2}(\Omega )^{d}}^{2} + \frac{{\mathbf{e}}^{C_{3}}}{2} \bigl\Vert \Pi _{ \mathcal{D}}B^{(0)} \bigr\Vert _{L^{2}(\Omega )}^{2}. \end{aligned} \end{aligned}$$
Taking the supremum over \(m\in [0,N]\) and using the real inequality \(\sup_{n}(r_{n}+s_{n}) \leq \sup_{n}(r_{n})+\sup {_{n}}(s_{n})\), we obtain the desired estimates. □
In the following definition, we introduce a dual norm [8], which is defined on the space \(\Pi _{\mathcal{D}}(X_{{\mathcal{D}},0}) \subset L^{2}(\Omega )\), to ensure the required compactness results.
Definition 4.3
If \({\mathcal{D}}\) is a gradient discretization, then the dual norm \(\| \cdot \|_{\star,{\mathcal{D}}}\) on \(\Pi _{\mathcal{D}}(X_{{\mathcal{D}},0})\) is given by
$$\begin{aligned} \begin{aligned}&\forall U\in \Pi _{\mathcal{D}}(X_{{\mathcal{D}},0}),\\ &\qquad \Vert U \Vert _{\star,{ \mathcal{D}}}=\sup \biggl\{ \int _{\Omega }U(\pmb{x})\Pi _{\mathcal{D}} \psi (\pmb{x}) \,\mathrm{d}\pmb{x}: \psi \in X_{{\mathcal{D}},0}, \Vert \nabla _{\mathcal{D}}\psi \Vert _{L^{2}(\Omega )^{d}} =1 \biggr\} . \end{aligned} \end{aligned}$$
(4.12)
Lemma 4.4
Under Hypothesis 2.1, let \({\mathcal{D}}\) be a gradient discretization, which is coercive. If \(B \in X_{{\mathcal{D}},0}\) satisfies (3.4b), then there exists a constant \(C_{4}\) depending only on \(C_{1}\), M, Ω, T, and \(\| \Pi _{\mathcal{D}}B^{(0)} \|_{L^{2}(\Omega )}\), such that
$$\begin{aligned} \int _{0}^{T} \bigl\Vert \delta _{\mathcal{D}}B(t) \bigr\Vert _{\star,{\mathcal{D}}}^{2} \,\mathrm{d}t \leq C_{4}. \end{aligned}$$
(4.13)
Proof
Putting \(\psi =\phi \) in (3.4b), together with the Cauchy–Schwarz inequality and the coercivity property, implies
$$\begin{aligned} \begin{aligned} & \int _{\Omega }\delta _{\mathcal{D}}^{(n+\frac{1}{2})} B(\pmb{x}) \Pi _{ \mathcal{D}}\phi (\pmb{x}) \,\mathrm{d}\pmb{x} \\ &\quad\leq d_{2} \bigl\Vert \nabla _{\mathcal{D}}B^{(n+1)} \bigr\Vert _{L^{2}(\Omega \times (0,T))^{d}} \Vert \nabla _{\mathcal{D}}\phi \Vert _{L^{2}(\Omega \times (0,T))^{d}} \\ &\qquad{} + \bigl( M \bigl\Vert \Pi _{\mathcal{D}}B^{(n+1)} \bigr\Vert _{L^{2}(\Omega \times (0,T))} + M \bigl\Vert \Pi _{\mathcal{D}}A^{(n+1)} \bigr\Vert _{L^{2}(\Omega \times (0,T))} + C_{0} \bigr) \Vert \Pi _{\mathcal{D}}\phi \Vert _{L^{2}( \Omega )} \\ &\quad\leq \Vert \nabla _{\mathcal{D}}\phi \Vert _{L^{2}(\Omega )^{d}} \bigl[ d_{2} \bigl\Vert \nabla _{\mathcal{D}}B^{(n+1)} \bigr\Vert _{L^{2}(\Omega \times (0,T))^{d}} \\ &\qquad{} +C_{\mathcal{D}} \bigl( M \bigl\Vert \Pi _{\mathcal{D}}B^{(n+1)} \bigr\Vert _{L^{2}( \Omega \times (0,T))} + M \bigl\Vert \Pi _{\mathcal{D}}A^{(n+1)} \bigr\Vert _{L^{2}( \Omega \times (0,T))} + C_{0} \bigr) \bigr]. \end{aligned} \end{aligned}$$
Taking the supremum over \(\phi \in X_{{\mathcal{D}},0}\) with \(\| \nabla _{\mathcal{D}}\phi \|_{L^{2}(\Omega )^{d}}=1\), multiplying by \(\delta t^{(n+1)}\), summing over \(n\in [0,N-1]\), and using (4.5) yield the desired estimate. □
Theorem 4.5
Under Hypothesis (2.1), let \(({\mathcal{D}}_{m})_{m\in {\mathbb{N}}}\) be a sequence of gradient discretizations, that is coercive, limit-conforming, consistent, compact, and such that \({\mathcal{K}}_{{\mathcal{D}}_{m}}\) is a nonempty set for any \(m\in {\mathbb{N}}\). For \(m \in {\mathbb{N}}\), let \((A_{m},B_{m}) \in {\mathcal{K}}_{{\mathcal{D}}_{m}}^{N_{m}+1} \times X_{{ \mathcal{D}}_{m},0}^{N_{m}+1}\) be solutions to the scheme (3.4a)–(3.4b) with \({\mathcal{D}}={\mathcal{D}}_{m}\). Then there exists a solution \((\bar{A},\bar{B})\) for the discrete problem (2.1a)–(2.1b), and a subsequence of gradient discretizations, indexed by \(({\mathcal{D}}_{m})_{m\in {\mathbb{N}}}\), such that, as \(m\to \infty \),
-
(1)
\(\Pi _{{\mathcal{D}}_{m}}A_{m}\to A\) and \(\Pi _{{\mathcal{D}}_{m}}B_{m}\to B\) strongly in \(L^{\infty }(0,T;L^{2}(\Omega ))\),
-
(2)
\(\nabla _{{\mathcal{D}}_{m}}A_{m}\to \nabla A\) and \(\nabla _{{\mathcal{D}}_{m}}B_{m}\to \nabla B\) strongly in \(L^{2}(\Omega \times (0,T))^{d}\),
-
(3)
\(\delta _{{\mathcal{D}}_{m}}A_{m}\) converges weakly to \(\partial _{t}\bar{A}\) in \(L^{2}(\Omega \times (0,T))\).
Proof
The proof is divided into four stages and its idea is inspired by [1].
Step 1: Existence of approximate solutions. At \((n+1)\), we see that (3.4a) and (3.4b) respectively express a system of nonlinear elliptic variational inequality on \(A^{(n+1)}\) and nonlinear equations on \(B^{(n+1)}\). For \(w=(w_{1},w_{2}) \in {\mathcal{K}}_{\mathcal{D}}\times X_{{\mathcal{D}},0}\), we see that \((A,B)\in {\mathcal{K}}_{\mathcal{D}}\times X_{{\mathcal{D}},0}\) satisfies
$$\begin{aligned} & a\bigl(A^{(n+1)},A^{(n+1)}-\varphi \bigr) \leq L \bigl(A^{(n+1)}-\varphi \bigr),\quad \forall \varphi \in {\mathcal{K}}_{\mathcal{D}}, \text{ and } \end{aligned}$$
(4.14a)
$$\begin{aligned} &\begin{aligned} {} &\int _{\Omega }\Pi _{{\mathcal{D}}} \frac{B^{(n+1)}-B^{(n)}}{\delta t^{(n+\frac{1}{2})}}(x) \Pi _{ \mathcal{D}}\psi (\pmb{x}) + \int _{\Omega }{\mathbf {D}_{B}}\nabla _{ \mathcal{D}}B^{(n+1)}( \pmb{x}) \cdot \nabla _{\mathcal{D}}\psi ( \pmb{x}) \,\mathrm{d}\pmb{x} \\ &\quad= \int _{\Omega }G(\Pi _{\mathcal{D}}w_{1}, \Pi _{{\mathcal{D}}} w_{2}) \Pi _{\mathcal{D}}\psi (\pmb{x}) \,\mathrm{d} \pmb{x}, \quad\forall \psi \in X_{{\mathcal{D}},0}, \end{aligned} \end{aligned}$$
(4.14b)
where \(\alpha:=\frac{1}{\delta t^{(n+\frac{1}{2})}}\) and the bilinear and linear forms are defined by
$$\begin{aligned} &a(\phi,z)=\alpha \int _{\Omega }\Pi _{\mathcal{D}}\phi \Pi _{ \mathcal{D}}z \,\mathrm{d}\pmb{x}+{\mathbf {D}_{A}} \int _{\Omega }\nabla _{ \mathcal{D}}\phi \cdot \nabla _{\mathcal{D}}z \,\mathrm{d}\pmb{x},\quad \forall \phi,z \in {\mathcal{K}}_{\mathcal{D}} \quad\text{and} \\ &L(z)= \int _{\Omega }F(\Pi _{\mathcal{D}}w_{1}, \Pi _{\mathcal{D}}w_{2}) \Pi _{\mathcal{D}}z \,\mathrm{d}\pmb{x}+ \alpha \int _{\Omega }\Pi _{ \mathcal{D}}A^{(n)} \Pi _{\mathcal{D}}z \,\mathrm{d}\pmb{x},\quad \forall z \in {\mathcal{K}}_{\mathcal{D}}. \end{aligned}$$
Stampacchia’s theorem implies that there exists \(\bar{A}\in {\mathcal{K}}_{\mathcal{D}}\) satisfying inequality (4.14a). The second equation (4.14b) describes a linear square system. Taking \(\varphi =B^{(n+1)}\) in (4.14b), using the similar reasoning as in the proof of Lemma 4.2, and setting \(G={\mathbf{0}}\) yield \(\| \nabla _{\mathcal{D}}B^{(n+1)} \|_{L^{2}(\Omega )^{d}}=0\). This shows that the matrix corresponding to the linear system is invertible. Consider the continuous mapping \(\mathbb{T}:{\mathcal{K}}_{\mathcal{D}}\times X_{{\mathcal{D}},0} \to { \mathcal{K}}_{\mathcal{D}}\times X_{{\mathcal{D}},0}\), where \(\mathbb{T}(w)=(A,B)\) with \((A,B)\) being the solution to (4.14a)–(4.14b). The existence of a solution \((A^{(n+1)},B^{(n+1)})\) to the nonlinear system is a consequence of Brouwer’s fixed point theorem.
Step 2: Strong convergence of \(\Pi _{{\mathcal{D}}_{m}}A_{m}\) and \(\Pi _{{\mathcal{D}}_{m}}B_{m}\) in \(L^{\infty }(0,T;L^{2}(\Omega ))\) and the weak convergence of \(\delta _{{\mathcal{D}}_{m}}A_{m}\) in \(L^{2}(\Omega \times (0,T))\). Applying estimate (4.5) to the sequence of solutions \(((A_{m})_{m\in {\mathbb{N}}},(B_{m})_{m\in {\mathbb{N}}})\) of the scheme (3.4a)–(3.4b) shows that both \(\| \nabla _{{\mathcal{D}}_{m}}A_{m} \|_{L^{2}(\Omega \times (0,T))^{d}}\) and \(\| \nabla _{{\mathcal{D}}_{m}}B_{m} \|_{L^{2}(\Omega \times (0,T))^{d}}\) are bounded. Using [8, Lemma 4.8], there exists a sequence, still denoted by \(({\mathcal{D}}_{m}^{T})_{m\in {\mathbb{N}}}\), and \(\bar{A}, \bar{B}\in L^{2}(0,T;H_{0}^{1}(\Omega ))\) such that, as \(m\to \infty \), \(\Pi _{{\mathcal{D}}_{m}}A_{m}\) converges weakly to Ā in \(L^{2}(\Omega \times (0,T))\), \(\nabla _{{\mathcal{D}}_{m}}A_{m}\) converges weakly to ∇Ā in \(L^{2}(\Omega \times (0,T))^{d}\), \(\Pi _{{\mathcal{D}}_{m}}B_{m}\) converges weakly to B̄ in \(L^{2}(\Omega \times (0,T))\), and \(\nabla _{{\mathcal{D}}_{m}}B_{m}\) converges weakly to ∇B̄ in \(L^{2}(\Omega \times (0,T))^{d}\). Since \(A_{m} \in {\mathcal{K}}_{{\mathcal{D}}_{m}}\), passing to the limit in \(\Pi _{{\mathcal{D}}_{m}}A_{m} \leq \chi \) in Ω shows that \(\bar{A} \leq \chi \) in Ω. Thanks to [8, Theorem 4.31], estimate (4.5) shows that \(\bar{A}\in C([0,T];L^{2}(\Omega ))\), \(\Pi _{{\mathcal{D}}_{m}}A_{m}\) converges strongly to Ā in \(L^{\infty }(0,T;L^{2}(\Omega ))\), and \(\delta _{{\mathcal{D}}_{m}}A_{m}\) converges weakly to \(\partial _{t} \bar{A}\) in \(L^{2}(0,T;L^{2}(\Omega ))\).
Let us show the strong convergence of \(\Pi _{{\mathcal{D}}_{m}}B_{m}\) to B̄ in \(L^{\infty }(0,T;L^{2}(\Omega ))\). Indeed, \(\Pi _{{\mathcal{D}}_{m}}B_{m}\) converges strongly to B̄ in \(L^{2}(\Omega \times (0,T))\), thanks to estimate (4.13), consistency, limit-conformity, and compactness, as well as [8, Theorem 4.14]. We can apply the dominated convergence theorem to show that \(G(\Pi _{{\mathcal{D}}_{m}}A_{m},\Pi _{{\mathcal{D}}_{m}}B_{m}) \to G( \bar{A},\bar{B})\) in \(L^{2}(\Omega \times (0,T))\), thanks to the assumptions on G given in Hypothesis 2.1.
Let \(t_{0} \in [0,T]\) and define the sequence \(t_{m} \in [0,T]\) such that \(t_{m} \to t_{0}\), as \(m \to \infty \). Consider \(k(m) \in [0,N_{m}-1]\) such that \(k_{m} \in (t^{(s(m))}, t^{(s(m)+1)}]\). Following the technique used in Lemma 4.2, one can obtain
$$\begin{aligned} \begin{aligned} &\frac{1}{2} \int _{\Omega }\bigl(\Pi _{{\mathcal{D}}_{m}} B(\pmb{x},t_{m}) \bigr)^{2} \,\mathrm{d}\pmb{x} \\ &\quad\leq \frac{1}{2} \int _{\Omega }\bigl(\Pi _{{\mathcal{D}}_{m}} J_{{\mathcal{D}}_{m}}B_{ \mathrm{ini}}( \pmb{x})\bigr)^{2} \,\mathrm{d}x - \int _{0}^{t^{(s(m))}} \int _{\Omega }{\mathbf {D}_{B}}\bigl(\nabla _{{\mathcal{D}}_{m}}B( \pmb{x},t)\bigr)^{2} \,\mathrm{d}\pmb{x} \,\mathrm{d}t \\ &\quad\leq \int _{0}^{t^{(s(m))}} \int _{\Omega }G\bigl(\Pi _{{\mathcal{D}}_{m}}A( \pmb{x},t),\Pi _{{\mathcal{D}}_{m}}B(\pmb{x},t)\bigr) \Pi _{{\mathcal{D}}_{m}}B( \pmb{x},t) \,\mathrm{d} \pmb{x} \,\mathrm{d}t. \end{aligned} \end{aligned}$$
(4.15)
Using the characteristic function, it is obvious that, as \(m \to \infty \),
$$\begin{aligned} \begin{aligned} &\Pi _{{\mathcal{D}}_{m}}B_{m} \to \bar{B} \quad\text{strongly in } L^{2}\bigl( \Omega \times (0,T)\bigr) \quad\text{and} \\ &{\mathbf{1}}_{[0,t^{(s(m))}]}\nabla \bar{B} \to {\mathbf{1}}_{[0,t_{0}]} \nabla \bar{B}\quad \text{strongly in } L^{2}\bigl(\Omega \times (0,T) \bigr)^{d}. \end{aligned} \end{aligned}$$
These convergence results imply that
$$\begin{aligned} & \int _{0}^{t_{0}} \int _{\Omega }{\mathbf {D}_{B}}\bigl(\nabla \bar{B}(\pmb{x},t) \bigr)^{2} \,\mathrm{d}\pmb{x} \,\mathrm{d}t \\ &\quad= \int _{0}^{t^{(s(m))}} \int _{\Omega }{\mathbf{1}}_{[0,s]}{\mathbf {D}_{B}} \bigl( \nabla \bar{B}(\pmb{x},t)\bigr)^{2} \,\mathrm{d}\pmb{x} \,\mathrm{d}t \\ &\quad=\lim_{m\to \infty } \int _{0}^{T} \int _{\Omega }{\mathbf{1}}_{[0,t^{(s(m))}]}{ \mathbf {D}_{B}} \nabla \bar{B}(\pmb{x},t) \cdot \nabla _{{\mathcal{D}}_{m}}B_{m}( \pmb{x},t) \,\mathrm{d}\pmb{x} \,\mathrm{d}t \\ &\quad\leq \liminf_{m\to \infty } \bigl( \Vert {\mathbf{1}}_{[0,t^{(s(m))}]} \nabla \bar{B} \Vert _{L^{2}(\Omega \times (0,T))^{d}} \cdot \Vert {\mathbf{1}}_{[0,t^{(s(m))}]}{ \mathbf {D}_{B}}\nabla _{{\mathcal{D}}_{m}}B_{m} \Vert _{L^{2}(\Omega \times (0,T))^{d}} \bigr) \\ &\quad= \Vert {\mathbf{1}}_{[0,t_{0}]}\nabla \bar{B} \Vert _{L^{2}(\Omega \times (0,T))^{d}} \cdot \liminf_{m\to \infty } \Vert {\mathbf{1}}_{[0,t^{(s(m))}]}{\mathbf {D}_{B}} \nabla _{{\mathcal{D}}_{m}}B_{m} \Vert _{L^{2}(\Omega \times (0,T))^{d}}. \end{aligned}$$
Dividing this inequality by \(\| {\mathbf{1}}_{[0,t_{0}]}\nabla \bar{B} \|_{L^{2}(\Omega \times (0,T))^{d}}\) gives
$$\begin{aligned} \begin{aligned} \int _{0}^{t_{0}} \int _{\Omega }{\mathbf {D}_{B}}\bigl(\nabla \bar{B}(\pmb{x},t) \bigr)^{2} \,\mathrm{d}\pmb{x} \,\mathrm{d}t \leq \liminf _{m\to \infty } \int _{0}^{t^{(s(m))}} \int _{\Omega }{\mathbf {D}_{B}}\bigl( \nabla _{{\mathcal{D}}_{m}}B_{m}(\pmb{x},t)\bigr)^{2} \,\mathrm{d}\pmb{x} \,\mathrm{d}t. \end{aligned} \end{aligned}$$
(4.16)
Passing to the limit superior in (4.15), we arrive at
$$\begin{aligned} \begin{aligned}& \limsup_{m\to \infty } \frac{1}{2} \int _{\Omega }\bigl(\Pi _{{\mathcal{D}}_{m}}B_{m}( \pmb{x},t_{m})\bigr)^{2} \,\mathrm{d}\pmb{x} \\ &\quad\leq \frac{1}{2} \int _{\Omega }B_{\mathrm{ini}}(\pmb{x})^{2} \,\mathrm{d} \pmb{x}- \int _{0}^{t_{0}} \int _{\Omega }{\mathbf {D}_{B}} \bigl(\nabla \bar{B}( \pmb{x},t)\bigr)^{2} \,\mathrm{d}\pmb{x} \,\mathrm{d}t \\ &\qquad{} + \int _{0}^{t_{0}} \int _{\Omega }G\bigl(\bar{A}(\pmb{x},t),\bar{B}( \pmb{x},t)\bigr) \bar{B}(\pmb{x},t) \,\mathrm{d}\pmb{x} \,\mathrm{d}t. \end{aligned} \end{aligned}$$
(4.17)
Letting \(\psi =\bar{B}{\mathbf{1}}_{[0,t_{0}]}(t)\) in (3.4b) and integrating by parts, we obtain
$$\begin{aligned} \begin{aligned} &\frac{1}{2} \int _{\Omega }\bigl(\bar{B}(\pmb{x},t_{0}) \bigr)^{2} \,\mathrm{d} \pmb{x}+ \int _{0}^{t_{0}} \int _{\Omega }{\mathbf {D}_{B}}\bigl(\nabla \bar{B}( \pmb{x},t)\bigr)^{2} \,\mathrm{d}\pmb{x} \,\mathrm{d}t \\ &\quad=\frac{1}{2} \int _{\Omega }B_{\mathrm{ini}}(\pmb{x})^{2} \,\mathrm{d} \pmb{x}+ \int _{0}^{t_{0}} \int _{\Omega }G\bigl(\bar{A}(\pmb{x},t),\bar{B}( \pmb{x},t)\bigr) \bar{B}(\pmb{x},t) \,\mathrm{d}\pmb{x} \,\mathrm{d}t. \end{aligned} \end{aligned}$$
(4.18)
From (4.17) and (4.18), we obtain
$$\begin{aligned} \limsup_{m\to \infty } \int _{\Omega }\bigl(\Pi _{{\mathcal{D}}_{m}}B_{m}( \pmb{x},t_{m})\bigr)^{2} \,\mathrm{d}\pmb{x}\leq \int _{\Omega }\bar{B}( \pmb{x},t_{0})^{2} \,\mathrm{d}\pmb{x}. \end{aligned}$$
(4.19)
Estimates (4.5) and (4.13), together with [8, Theorem 4.19], imply the weak convergence of \((\Pi _{{\mathcal{D}}_{m}}B_{m})_{m\in {\mathbb{N}}}\) to B̄ in \(L^{2}(\Omega )\); it is indeed uniform in \([0,T]\). This yields the weak convergence of \(\Pi _{{\mathcal{D}}_{m}}B_{m}(\cdot,s_{m})\) to \(\bar{B}(\cdot,t_{0})\) in \(L^{2}(\Omega )\). As a consequence of estimate (4.19), this convergence of \(\Pi _{{\mathcal{D}}_{m}}B_{m}(\cdot,s_{m})\) holds in the strong sense in \(L^{2}(\Omega )\). With the continuity of \(\bar{B}:[0,T] \to L^{2}(\Omega )\), we can apply [8, Lemma C.13] to conclude the strong convergence of \(\Pi _{{\mathcal{D}}_{m}}B_{m}\) in \(L^{\infty }(0,T;L^{2}(\Omega ))\).
Step 3: Convergence towards the solution of the continuous model. Recall that \(A_{m}^{(0)}=I_{{\mathcal{D}}_{m}}\bar{A}_{\mathrm{ini}}\), therefore the consistency shows that \(\Pi _{{\mathcal{D}}_{m}}A_{m}^{(0)}\) converges strongly to \(A_{\mathrm{ini}}\) in \(L^{2}(\Omega )\), as \(m \to \infty \). Hence, \(\bar{A}\in C([0,T];L^{2}(\Omega ))\cap \mathbb{K}\) and Ā satisfies all the conditions except for the integral inequality imposed on the exact solution of problem (2.1a). Let us now show that this integral relation holds. With Hypothesis 2.1, the dominated convergence theorem leads to \(F(\Pi _{{\mathcal{D}}_{m}}A_{m},\Pi _{{\mathcal{D}}_{m}}B_{m}) \to F( \bar{A},\bar{B})\) in \(L^{2}(\Omega \times (0,T))\). The \(L^{2}\)-weak convergence of \(\nabla _{{\mathcal{D}}_{m}}A_{m}\) yields
$$\begin{aligned} \int _{0}^{T} \int _{\Omega }{\mathbf {D}_{A}}\nabla \bar{A}\cdot \nabla \bar{A} \,\mathrm{d}\pmb{x} \,\mathrm{d}t \leq \liminf_{m\to \infty } \int _{0}^{T} \int _{\Omega }{\mathbf {D}_{A}}\nabla _{{\mathcal{D}}_{m}}A_{m} \cdot \nabla _{{ \mathcal{D}}_{m}} A_{m} \,\mathrm{d}\pmb{x} \,\mathrm{d}t. \end{aligned}$$
(4.20)
Fix \(\kappa >0\) and let \(\bar{w}_{\kappa }\in \mathbb{L}_{\kappa }\), where \(\mathbb{L}_{\kappa }\) is defined by (4.1). Thanks to Lemma 4.1, there exists a sequence \((w_{m})_{m\in {\mathbb{N}}}\) such that \(w_{m} \in {\mathcal{K}}_{{\mathcal{D}}_{m}}^{N_{m}+1}\), and \(\Pi _{{\mathcal{D}}_{m}}w_{m} \to \bar{w}_{\kappa }\) strongly in \(L^{2}(\Omega \times (0,T))\) and \(\nabla _{{\mathcal{D}}_{m}}w_{m} \to \nabla \bar{w}_{\kappa }\) strongly in \(L^{2}(\Omega \times (0,T))^{d}\). Setting \(\varphi:=w_{m}\) as a generic function in (3.4a), inequality (4.20) implies that
$$\begin{aligned} & \int _{0}^{T} \int _{\Omega }{\mathbf {D}_{A}}\nabla \bar{A}\cdot \nabla \bar{A} \,\mathrm{d}\pmb{x} \,\mathrm{d}t \\ &\quad\leq \liminf_{m\to \infty } \biggl[ \int _{0}^{T} \int _{\Omega }F(\Pi _{{ \mathcal{D}}_{m}}A_{m},\Pi _{{\mathcal{D}}_{m}}B_{m}) \Pi _{{\mathcal{D}}_{m}}(A_{m}-w_{m}) \,\mathrm{d}\pmb{x} \,\mathrm{d}t \\ &\qquad{} + \int _{0}^{T} \int _{\Omega }{\mathbf {D}_{A}}\nabla _{{\mathcal{D}}_{m}}A_{m} \cdot \nabla _{{\mathcal{D}}_{m}} w_{m} \,\mathrm{d}\pmb{x} \,\mathrm{d}t- \int _{0}^{T} \int _{\Omega }\delta _{{\mathcal{D}}_{m}}A_{m} \Pi _{{\mathcal{D}}_{m}}(A_{m}-w_{m}) \,\mathrm{d}\pmb{x} \,\mathrm{d}t \biggr]. \end{aligned}$$
With the weak–strong convergences, we pass to the limit in this relation to obtain, for all \(w_{\kappa }\in \mathbb{L}_{\kappa }\) and all \(\kappa >0\),
$$\begin{aligned} \begin{aligned} {} &\int _{0}^{T} \int _{\Omega }\partial _{t} \bar{A}(\pmb{x},t) (u-w_{\kappa }) (\pmb{x},t) \,\mathrm{d}\pmb{x} \,\mathrm{d}t + \int _{0}^{T} \int _{\Omega }{\mathbf {D}_{A}}\nabla \bar{A} \cdot \nabla ( \bar{A}-w_{\kappa }) ( \pmb{x},t) \,\mathrm{d}\pmb{x} \,\mathrm{d}t \\ &\quad\leq \int _{0}^{T} \int _{\Omega }F\bigl(\bar{A}(\pmb{x},t),\bar{B}(\pmb{x},t)\bigr) ( \bar{A}-w_{\kappa }) (\pmb{x},t) \,\mathrm{d}\pmb{x} \,\mathrm{d}t. \end{aligned} \end{aligned}$$
By the density of the set \(C_{0}^{\infty }(\overline{\Omega })\cap {\mathcal{K}}\) in \({\mathcal{K}}\) proved in [14], every \(\varphi \in \mathbb{K}\) can be approximated by a piecewise constant function in time \(w_{\kappa }\in \mathbb{L}_{\kappa }\) such that \(w_{\kappa }\to \varphi \) strongly in \(L^{2}(0,T;H_{0}^{1}(\Omega ))\) as \(\kappa \to 0\) (note that \(w_{\kappa }\leq \chi \) in \(\Omega \times (0,T)\)). Hence, (2.1a) is verified.
Let us verify the integral equality (2.1b). Let ψ be a generic function in the space \(L^{2}(0,T;L^{2}(\Omega ))\) which satisfies \(\partial _{t} \psi \in L^{2}(\Omega \times (0,T))\) and \(\psi (T,\cdot )=0\). Using the technique in [8, Lemma 4.10], we can construct \(w_{m}=(w_{m}^{(n)})_{n=0,\ldots,N_{m}} \in X_{{\mathcal{D}}_{m},0}^{N_{m}+1}\), such that \(\Pi _{{\mathcal{D}}_{m}}w_{m} \to \psi \) in \(L^{2}(0,T;L^{2}(\Omega ))\) and \(\delta _{{\mathcal{D}}_{m}}w_{m} \to \partial _{t} \psi \) strongly in \(L^{2}(\Omega \times (0,T))\). Take \(\psi =\delta t_{m} ^{(n+\frac{1}{2})}w_{m}^{(n)}\) as a generic function in (3.4b) and sum over \(n\in [0,N_{m}-1]\) to get
$$\begin{aligned} \begin{aligned} &\sum_{n=0}^{N_{m}-1} \int _{\Omega }\bigl[ \Pi _{{\mathcal{D}}_{m}}B_{m}^{(n+1)}( \pmb{x})-\Pi _{{\mathcal{D}}_{m}}B^{(n)}(\pmb{x}) \bigr]\Pi _{{ \mathcal{D}}_{m}}w_{m}^{(n)}(\pmb{x}) \,\mathrm{d}x \\ &\qquad{}+ \int _{0}^{T} \int _{\Omega }{\mathbf {D}_{B}}(\pmb{x})\nabla _{{\mathcal{D}}_{m}}B_{m}( \pmb{x},t) \cdot \nabla _{{\mathcal{D}}_{m}}w_{m}( \pmb{x},t) \,\mathrm{d}\pmb{x} \,\mathrm{d}t \\ &\quad= \int _{0}^{T} \int _{\Omega }G\bigl(\Pi _{{\mathcal{D}}_{m}}A(\pmb{x},t), \Pi _{{\mathcal{D}}_{m}}B(\pmb{x},t)\bigr)\Pi _{{\mathcal{D}}_{m}}w_{m}( \pmb{x},t) \,\mathrm{d}x \,\mathrm{d}t. \end{aligned} \end{aligned}$$
(4.21)
Applying [8, Eq. (D.15)] to the right-hand side yields, thanks to \(w^{(N)}=0\),
$$\begin{aligned} \begin{aligned} &{-} \int _{0}^{T} \int _{\Omega }\Pi _{{\mathcal{D}}_{m}}B_{m}(\pmb{x},t) \delta _{{\mathcal{D}}_{m}}w_{m}(\pmb{x},t) \,\mathrm{d}\pmb{x} \,\mathrm{d}t - \int _{\Omega }\Pi _{{\mathcal{D}}_{m}}B_{m}^{(0)}( \pmb{x})\Pi _{{\mathcal{D}}_{m}}w_{m}^{(0)}(\pmb{x}) \,\mathrm{d} \pmb{x} \\ &\qquad{}+ \int _{0}^{T} \int _{\Omega }{\mathbf {D}_{B}}(\pmb{x})\nabla _{{\mathcal{D}}_{m}}B_{m}( \pmb{x},t) \cdot \nabla _{{\mathcal{D}}_{m}}w_{m}( \pmb{x},t) \,\mathrm{d}\pmb{x} \,\mathrm{d}t \\ &\quad= \int _{0}^{T} \int _{\Omega }G\bigl(\Pi _{{\mathcal{D}}_{m}}A(\pmb{x},t), \Pi _{{\mathcal{D}}_{m}}B(\pmb{x},t)\bigr)\Pi _{{\mathcal{D}}_{m}}w_{m}( \pmb{x},t) \,\mathrm{d}x \,\mathrm{d}t. \end{aligned} \end{aligned}$$
Using the consistency, we see that \(\Pi _{{\mathcal{D}}_{m}}B_{m}^{(0)}=\Pi _{{\mathcal{D}}_{m}}J_{{ \mathcal{D}}_{m}}B_{\mathrm{ini}} \to B_{\mathrm{ini}}\) in \(L^{2}(\Omega )\). This, when passing to the limit \(m\to \infty \), implies, for all ψ in \(L^{2}(0,T;H^{1}(\Omega ))\),
$$\begin{aligned} \begin{aligned} &{-} \int _{0}^{T} \int _{\Omega }\partial _{t} \psi (\pmb{x},t)\bar{B}( \pmb{x},t) \,\mathrm{d}\pmb{x} \,\mathrm{d}t + \int _{0}^{T} \int _{\Omega }{\mathbf {D}_{B}}\nabla \bar{B} \cdot \nabla \psi (\pmb{x},t) \,\mathrm{d}\pmb{x} \,\mathrm{d}t \\ &\quad= \int _{0}^{T} \int _{\Omega }G(\bar{A},\bar{B})\psi (\pmb{x},t) \,\mathrm{d}\pmb{x} \,\mathrm{d}t. \end{aligned} \end{aligned}$$
Since \(C^{\infty }([0,T];H^{1}(\Omega ))\) is dense in \(L^{2}(0,T;H^{1}(\Omega ))\), integrating by parts shows that the above equality can be expressed in the sense of distributions, which is equivalent to (2.1b).
Step 4: Proof of the strong convergence of \(\nabla _{{\mathcal{D}}_{m}}A_{m}\) and \(\nabla _{{\mathcal{D}}_{m}}B_{m}\). From the weak–strong convergences, we have, for all \(\bar{w}_{\kappa }\in \mathbb{L}_{\kappa }\),
$$\begin{aligned} &\limsup_{m\to \infty } \int _{0}^{T} \int _{\Omega }{\mathbf {D}_{A}}\nabla _{{ \mathcal{D}}_{m}}A_{m} \cdot \nabla _{{\mathcal{D}}_{m}} A_{m} \,\mathrm{d}\pmb{x} \,\mathrm{d}t \\ &\quad\leq \int _{0}^{T} \int _{\Omega }F(\bar{A},\bar{B}) (\bar{A}-\bar{w}_{\kappa }) \,\mathrm{d}\pmb{x} \,\mathrm{d}t \\ &\qquad{} + \int _{0}^{T} \int _{\Omega }{\mathbf {D}_{A}}\nabla \bar{A}\cdot \nabla \bar{w}_{\kappa }\,\mathrm{d}\pmb{x} \,\mathrm{d}t - \int _{0}^{T} \int _{\Omega }\partial _{t} \bar{A}(\bar{A}- \bar{w}_{\kappa }) \,\mathrm{d} \pmb{x} \,\mathrm{d}t. \end{aligned}$$
Thanks to the density results, for any \(\varphi \in \mathbb{K}\), we can find \((\bar{w}_{\kappa })_{\kappa >0}\) that converges to φ in \(L^{2}(0,T;H^{1}(\Omega ))\), as \(\kappa \to 0\). Therefore, we infer, for all \(\varphi \in \mathbb{K}\),
$$\begin{aligned} &\limsup_{m\to \infty } \int _{0}^{T} \int _{\Omega }{\mathbf {D}_{A}}\nabla _{{ \mathcal{D}}_{m}}A_{m} \cdot \nabla _{{\mathcal{D}}_{m}} A_{m} \,\mathrm{d}\pmb{x} \,\mathrm{d}t \\ &\quad\leq \int _{0}^{T} \int _{\Omega }F(\bar{A},\bar{B}) (\bar{A}-\varphi ) \,\mathrm{d} \pmb{x} \,\mathrm{d}t \\ & \qquad{}+ \int _{0}^{T} \int _{\Omega }{\mathbf {D}_{A}}\nabla \bar{A}\cdot \nabla \varphi \,\mathrm{d}\pmb{x} \,\mathrm{d}t - \int _{0}^{T} \int _{\Omega }\partial _{t} \bar{A}(\bar{A}-\varphi ) \,\mathrm{d}\pmb{x} \,\mathrm{d}t. \end{aligned}$$
Taking \(\varphi =\bar{A}\), the above relation yields
$$\begin{aligned} \limsup_{m\to \infty } \int _{0}^{T} \int _{\Omega }{\mathbf {D}_{A}}\nabla _{{ \mathcal{D}}_{m}}A_{m} \cdot \nabla _{{\mathcal{D}}_{m}} A_{m} \,\mathrm{d}\pmb{x} \,\mathrm{d}t \leq \int _{0}^{T} \int _{\Omega }{\mathbf {D}_{A}} \nabla \bar{A}\cdot \nabla \bar{A} \,\mathrm{d}\pmb{x} \,\mathrm{d}t. \end{aligned}$$
Together with (4.20), we obtain
$$\begin{aligned} \lim_{m\to \infty } \int _{0}^{T} \int _{\Omega }{\mathbf {D}_{A}}\nabla _{{ \mathcal{D}}_{m}}A_{m} \cdot \nabla _{{\mathcal{D}}_{m}} A_{m} \,\mathrm{d}\pmb{x} \,\mathrm{d}t = \int _{0}^{T} \int _{\Omega }{\mathbf {D}_{A}} \nabla \bar{A}\cdot \nabla \bar{A} \,\mathrm{d}\pmb{x} \,\mathrm{d}t, \end{aligned}$$
which implies
$$\begin{aligned} 0\leq{}& d_{1}\limsup_{m\to \infty } \int _{0}^{T} \int _{\Omega } \vert \nabla \bar{A}-\nabla _{{\mathcal{D}}_{m}} A_{m} \vert ^{2} \,\mathrm{d}\pmb{x} \,\mathrm{d}t \\ \leq{}& \limsup_{m\to \infty } \biggl[ \int _{0}^{T} \int _{\Omega }{\mathbf {D}_{A}} \nabla \bar{A} \cdot \nabla \bar{A} + \int _{0}^{T} \int _{\Omega }{\mathbf {D}_{A}} \nabla _{{\mathcal{D}}_{m}}A_{m} \cdot \nabla _{{\mathcal{D}}_{m}}A_{m} \,\mathrm{d}\pmb{x} \,\mathrm{d}t \\ &{} -2 \int _{0}^{T} \int _{\Omega }{\mathbf {D}_{A}}\nabla \bar{A} \cdot \nabla _{{\mathcal{D}}_{m}}A_{m} \,\mathrm{d}\pmb{x} \,\mathrm{d}t \biggr] =0, \end{aligned}$$
showing that \(\nabla _{{\mathcal{D}}_{m}}A_{m} \to \nabla \bar{A}\) strongly in \(L^{2}(\Omega \times (0,T))^{d}\). To show the strong convergence of \(\nabla _{{\mathcal{D}}_{m}}B_{m}\), we begin by writing
$$\begin{aligned} \begin{aligned} &\int _{0}^{T} \int _{\Omega }\bigl(\nabla _{{\mathcal{D}}_{m}}B_{m}( \pmb{x},t)- \nabla \bar{B}(\pmb{x},t)\bigr)\cdot \bigl(\nabla _{{\mathcal{D}}_{m}}B_{m}( \pmb{x},t)- \nabla \bar{B}(\pmb{x},t)\bigr) \,\mathrm{d}\pmb{x} \,\mathrm{d}t \\ &\quad= \int _{0}^{T} \int _{\Omega }\nabla _{{\mathcal{D}}_{m}}B_{m}(\pmb{x},t) \cdot \nabla _{{\mathcal{D}}_{m}}B_{m}(\pmb{x},t) \,\mathrm{d}\pmb{x} \,\mathrm{d}t \\ &\qquad{} - \int _{0}^{T} \int _{\Omega }\nabla _{{\mathcal{D}}_{m}}B_{m}( \pmb{x},t) \cdot \nabla \bar{B}(\pmb{x},t) \,\mathrm{d}\pmb{x} \,\mathrm{d}t \\ &\qquad{} - \int _{0}^{T} \int _{\Omega }\nabla \bar{B}(\pmb{x},t) \cdot \bigl( \nabla _{{\mathcal{D}}_{m}}B_{m}(\pmb{x},t)-\nabla \bar{B}(\pmb{x},t)\bigr) \,\mathrm{d}\pmb{x} \,\mathrm{d}t. \end{aligned} \end{aligned}$$
(4.22)
Setting \(\psi:=B_{m}\) in (3.4b) and \(\psi = \bar{B}\) in (2.1b), and taking \({\mathbf {D}_{B}}(\pmb{x})={\mathbf{Id}}\), when passing to the limit superior, yields
$$\begin{aligned} \begin{aligned} &\limsup_{m\to \infty } \int _{0}^{T} \int _{\Omega }\nabla _{{\mathcal{D}}_{m}}B_{m}( \pmb{x},t) \cdot \nabla _{{\mathcal{D}}_{m}}B_{m}(\pmb{x},t) \,\mathrm{d}x \,\mathrm{d}t \\ &\quad= \int _{0}^{T} \int _{\Omega }G(\bar{A},\bar{B})\bar{B}(\pmb{x},t) \,\mathrm{d} \pmb{x} \,\mathrm{d}t - \int _{0}^{T} \int _{\Omega }\partial _{t} \bar{B}(\pmb{x},t) \bar{B}( \pmb{x},t) \,\mathrm{d}\pmb{x} \,\mathrm{d}t \\ &\quad= \int _{0}^{T} \int _{\Omega }\nabla \bar{B}(\pmb{x},t)\cdot \nabla \bar{B}( \pmb{x},t) \,\mathrm{d}\pmb{x} \,\mathrm{d}t. \end{aligned} \end{aligned}$$
Passing to the limit in (4.22) and using the above inequality, we reach the desired convergence result. □