I am doing a past qualify exam question.
Let $$W_{h}=\left\{v_{h} \in \mathcal{C}^{0}(\bar{\Omega}) ; \forall \tau \in \mathcal{T}_{h},\left.v_{h}\right|_{\tau} \in \mathcal{P}_{1},\left.v_{h}\right|_{\partial \Omega}=0\right\}$$ Consider the fully discrete backward Euler implicit scheme: for $K$ a positive integer, set $k=T / K,$ define $t_{n}=n k, 0 \leq n \leq K,$ and for each $0 \leq n \leq K-1,$ knowing $u_{h}^{n} \in W_{h}$ find $u_{h}^{n+1} \in W_{h}$ such that
$$ \forall v_{h} \in W_{h}, \frac{1}{k}\left(u_{h}^{n+1}-u_{h}^{n}, v_{h}\right)+a\left(u_{h}^{n+1}, v_{h}\right)=0, \\n=0,1, \cdots, K,~~ u_{h}^{0}=\Pi_{h}\left(u_{0}\right) $$ where $a\left(u_{h}^{n+1}, v_{h}\right)=\int_{\Omega} \nabla u_{h}^{n+1} \cdot \nabla v_{h}+c u_{h}^{n+1} v_{h} d x$ , $c>0$ is a constant.
Please prove the following stability estimate
(1) $$ \sup _{1 \leq n \leq K}\left\|u_{h}^{n}\right\|_{L^{2}(\Omega)}^{2}+k \sum_{n=1}^{K}\left|u_{h}^{n}\right|_{H^{1}(\Omega)}^{2} \leq\left\|u_{h}^{0}\right\|_{L^{2}(\Omega)}^{2} $$
(2)Also prove the estimate $$ \sup _{1 \leq n \leq K}\left|u_{h}^{n}\right|_{H^{1}(\Omega)} \leq\left|u_{h}^{0}\right|_{H^{1}(\Omega)} $$
For the first question, I have tried to let $v_h=u_h^{n+1}$ but could not get this specific estimate. For the second question, I haven't any idea.
Here is my try for the first question.
