## 3 Numerical Scheme
### Remarks on Existence and Convergence Theory to the Numerical Scheme
#### 3.1.2 Notes on the Convergence Theory
For the proof of the following Lemma, recall the \(\Gamma\)-convergence. As defined in [36; 37; 38; 39], a sequence of nonlinear functionals \(F_{n}:X\to\mathbb{R}\) is said to \(\Gamma\)-converge to a functional \(F:X\to\mathbb{R}\) if the following two conditions are fulfilled:  
(i) For every sequence \((x_{n})\subset X\), with \(x=\lim\limits_{n\to\infty}x_{n}\), the following inequality holds true:  
\[F(x)\leq\liminf_{n\to\infty}F_{n}(x_{n}) \tag{24}\]  
(ii) For every \(x\in X\), there exists a sequence \((x_{n})\subset X\) with \(x=\lim\limits_{n\to\infty}x_{n}\), such that the following inequality holds true:  
\[F(x)\geq\limsup_{n\to\infty}F_{n}(x_{n}). \tag{25}\]  
**Lemma 1**. : _Let \(\mathcal{T}_{n}\) be a sequence of triangulations with \(h_{n}\downarrow 0\), where \(h_{n}\) denotes the diameter of \(\mathcal{T}_{n}\), which fulfills \(0=\lim\limits_{n\to\infty}h_{n}\), as well as that \(\mathcal{T}_{n+1}\) is a refinement of \(\mathcal{T}_{n}\), i.e., the set of nodes \(\mathcal{N}_{n}\) of \(\mathcal{T}_{n}\) is included in the set of nodes \(\mathcal{N}_{n+1}\) of \(\mathcal{T}_{n+1}\), i.e., \(\mathcal{N}_{n}\subset\mathcal{N}_{n+1}\), then the sequence \(F_{n}:X\to\mathbb{R}\) given through,_  
\[F_{n}:=F+\delta_{T_{n}},\]  
\(\Gamma\)_-converges to \(F:=LS_{x_{j}}(x,\lambda^{(j)})\). In the notation above, \(\delta_{n}\) is given through the following definition:_  
\[\delta_{n}(x)= \begin{cases}0,&\text{for }x\in X_{h_{n}},\\ \infty,&\text{else}.\end{cases} \]Proof. : To prove the \(\Gamma\)-convergence of the sequence \((F_{n})\), the assertions (24) and (25) have to be proven. 1. _ad_ (24): for the proof of this claim one needs to distinguish two cases. However, first, let \(x\in X\) and \(x^{(k)}\subset X\) with \(x=\lim_{k\to\infty}x^{(k)}\) be arbitrary fixed. 1. First, assume that up to finitely many \(n\in\mathbb{N}\) one has \(\
eg(x_{n}\in X_{h_{n}})\). From the definition of \(F_{n}\) it follows then that, up to finitely many \(n\in\mathbb{N}\), the lowest accumulation point of the sequence \(F_{n}(x_{n})=+\infty\). Hence, one obtains the following inequality: \[F(x)\leq\liminf_{n\to\infty}F_{n}(x_{n})=\infty.\] The inequality above shows the assertion for this case. 2. For the second case, assume that there are infinitely many \(n\in\mathbb{N}\), such that \(x_{n}\in X_{h_{n}}\). Subsequently, there exists a subsequence \((x_{n_{l}})\subset(x_{n})\) with \[F_{n_{l}}(x_{n_{l}})=F(x_{n_{l}}).\] Because of the continuity of \(F\) and the definition of \(F_{n}\), it follows: \[F(x)=\lim_{l\to\infty}F(x_{n_{l}})=\lim_{l\to\infty}F_{n_{l}}(x_{n_{l}})=\liminf _{n\to\infty}F_{n}(x_{n}).\] The equalities above yield that the claim for the second case holds true. The discussion of the two cases above proves the claim. 2. _ad_ (25): first, note that, for all \(n\in\mathbb{N}\), one has a continuous embedding \(S^{1}(\mathcal{T})\times\hookrightarrow W^{k,2}(\mathcal{T})\) and \(\bigcup\limits_{n\to\infty}S^{1}(\mathcal{T}_{n})\) is a dense subset of \(W^{k,2}(\Omega)\), c.f. [19; 27; 28; 29]. Similarly, one has a continuous embedding \(RT_{0}(\mathcal{T})\hookrightarrow H(\operatorname{div},\Omega)\) furthermore \(\bigcup\limits_{n\to\infty}RT_{0}(\mathcal{T})\) is a dense subset of \(H(\operatorname{div},\Omega)\), c.f. [19; 27; 28; 29]. From the definitions of \(X_{h}\) and \(X\), it follows with the both densities, where \(\bigcup\limits_{n\to\infty}X_{h_{n}}\) is a dense subset of \(X\). From the assumptions on the sequence \(\mathcal{T}_{n}\) and the density of \(\bigcup\limits_{n\to\infty}\) it follows that there exist, for all \(x\in X\), a sequence with \(x_{n}\in X_{h_{n}}\) s.t. \(x=\lim\limits_{n\to\infty}x_{n}\). Furthermore, it follows from the continuity of \(F\) and the definitions above: \[F(x)=\lim\limits_{n\to\infty}F(x_{n})=\lim\limits_{n\to\infty}F_{n}(x_{n})= \limsup\limits_{n\to\infty}F_{n}(x_{n}).\] (26) The equalities above prove the statement. From (i) and (ii), the claim of the Lemma follows by definition. **Corollary 1**.