Let $V_h \subset \mathrm{H}'_0(\omega)$. $V_h$ is finite dimensional. The finite element with Crank-Nicolson for solving $$ \left\{ \begin{array}{ccc}{\frac{\partial P}{\partial t}-\nabla \cdot\left(\underline{a} \nabla_{1}\right)=0} & {\text{on } \Omega \times(0,T]} \\ {P=0} & {\text{on } \partial\Omega \times(0,T]}\\ {P(\cdot, 0) = P_{0}} & {\text{in }\Omega}\end{array} \right. $$ is $(\underline{a}>0)$. How can we find $P_h^n\in V_n$ such that $$ \left\{ \begin{array}{ccc}{(\frac{P_{n}^{h}-P_{h}^{n-1}}{\Delta t}, v)+a(\frac{P_{n}^{h}+P_{h}^{n-1}}{\Delta t},v)=0} & {\forall v\in V_h} \\ {\left(P_{h}^{0}, v\right)=\left(P_{0}, V\right)} & {\forall v\in V_h}\end{array} \right. $$ where $v = \frac{P_{n}^{h}+P_{h}^{n-1}}{2}$ and $a(p,v)=\int_{\Omega}\underline{a}\nabla p\cdot\nabla v dx$. How can we show this is unconditionally unstable?


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