## 3 Numerical Scheme
In this paper, the estimator_ \(\sigma:\mathcal{T}\to\mathbb{R}_{\geq 0}\) _and the discrete solution of the discrete solution from_ _S_**1:** _of Algorithm_ 3_, for_ \(\alpha=0\) _and_ \(\mu\equiv 0\)_, is defined as_ \[\eta^{2}(T) :=\big{\|}D_{A}\mathrm{div}q_{A}+\gamma_{A}c_{A}+k_{2}v_{C}-k_{1}v _{A}v_{B}+f_{A}\big{\|}_{L^{2}(T)}^{2}+\big{\|}q_{A}-\
abla v_{A}\big{\|}_{L^{2 }(T)}^{2}\] \[+\big{\|}D_{B}\mathrm{div}q_{B}+\gamma_{B}g_{B}+k_{2}v_{C}-k_{1}v _{A}v_{B}+f_{B}\big{\|}_{L^{2}(T)}^{2}+\big{\|}q_{B}-\
abla v_{B}\big{\|}_{L^{2 }(T)}^{2}\] \[+\big{\|}D_{C}\mathrm{div}q_{C}+\gamma_{C}c_{C}-k_{2}v_{C}+k_{1}v _{A}v_{B}+f_{C}\big{\|}_{L^{2}(T)}^{2}+\big{\|}q_{C}-\
abla v_{C}\big{\|}_{L^{2 }(T)}^{2}\] \[+\sum_{S\in\{A,B,C\}}\frac{\alpha}{2}\big{\|}(\lambda_{S}-v_{S})_ {+}\big{\|}_{L^{2}(T)}^{2}\] \[\sum_{S\in\{A,B,C\}}\left[\left\|c^{j,n}_{S,h}-c^{j,n-1}_{S,h}\right\|^{2}_{L ^{2}(\Omega)}+\left\|q^{j,n}_{S,h}-q^{j,n-1}_{S,h}\right\|^{2}_{L^{2}(\Omega)} \right]<\varepsilon\]0
2. _Because, for the solution_ \((u_{LS,S}^{(j)},p_{LS,S}^{(j)})\in X\) _of (_5_), in the iteration_ \(j\in\mathbb{N}\)_, for given_ \(\alpha_{j}\) _and_ \(\mu^{(j)}\)_, one cannot expect the identity_ \[\sum_{S\in\{A,B,C\}}\left[\left\|c^{j,n}_{S,h}-c^{j,n-1}_{S,h}\right\|^{2}_{L ^{2}(\Omega)}+\left\|q^{j,n}_{S,h}-q^{j,n-1}_{S,h}\right\|^{2}_{L^{2}(\Omega)} \right]<\varepsilon\]1 (16) _to hold true, since a solution of (_5_) is in general not a solution to a system of non-linear equations. Hence, the usage of_ \[\sum_{S\in\{A,B,C\}}\left[\left\|c^{j,n}_{S,h}-c^{j,n-1}_{S,h}\right\|^{2}_{L ^{2}(\Omega)}+\left\|q^{j,n}_{S,h}-q^{j,n-1}_{S,h}\right\|^{2}_{L^{2}(\Omega)} \right]<\varepsilon\]2 _for the solution_ \((u_{LS,S,k^{\prime}}^{(j)},p_{LS,S,h}^{(j)})_{S\in\{A,B,C\}}\) _of the discrete problem of_ \(S\) **1** _of Algorithm_ 3_, is not practical. In the spirit of the usage of adaptive schemes an estimator_ \(\tau\)_, basing on_ \(\rho\)_, is used, which gives an estimation on the local changes w.r.t. the elements_ \(T\in\mathcal{T}\) _of the evaluation of_ \(\rho\)_. For the definition of a practical estimator let for an arbitrary fixed_ \(T\in\mathcal{T}\) _the element patch_ \(\mathcal{T}(T)\) _be given through_ \[\sum_{S\in\{A,B,C\}}\left[\left\|c^{j,n}_{S,h}-c^{j,n-1}_{S,h}\right\|^{2}_{L ^{2}(\Omega)}+\left\|q^{j,n}_{S,h}-q^{j,n-1}_{S,h}\right\|^{2}_{L^{2}(\Omega)} \right]<\varepsilon\]3 _Afterwards, the new estimator_ \(\sigma\) _is given through:_ \[\sum_{S\in\{A,B,C\}}\left[\left\|c^{j,n}_{S,h}-c^{j,n-1}_{S,h}\right\|^{2}_{L ^{2}(\Omega)}+\left\|q^{j,n}_{S,h}-q^{j,n-1}_{S,h}\right\|^{2}_{L^{2}(\Omega)} \right]<\varepsilon\]4 (17)
3. _A common strategy, used for the marking is the so-called Dorfler marking, cf._ _[_49_]__. In this strategy, a set_ \(\mathcal{M}\) _is chosen for which,_ \(\tau^{2}\in\{\eta^{2},\sigma^{2},\rho^{2}\}\)_, for a given bulk parameter_ \(\theta\in[0,1]\) _the inequality_ \[\sum_{S\in\{A,B,C\}}\left[\left\|c^{j,n}_{S,h}-c^{j,n-1}_{S,h}\right\|^{2}_{L ^{2}(\Omega)}+\left\|q^{j,n}_{S,h}-q^{j,n-1}_{S,h}\right\|^{2}_{L^{2}(\Omega)} \right]<\varepsilon\]5 (18) _holds true._ _Note that the Dorfler marking implies for_ \(\theta=1\) _a simple uniform marking scheme._ _As discussed in_ _[_50; 51_]__, one can usually prove convergence of the adaptive scheme by using, as in this article, the least-squares functional as error estimator, but cannot expect optimal convergence rates as discussed in [24] or [47]. Hence, another error estimator has to be derived for which optimal convergence rates are provable._  
With the remark and definition above, the numerical scheme is completed. The numerical validation will be made in Section 4, the numerical scheme will be validated.