# Stability of FEM using Crank-Nicolson

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?