= Solution
{c}
Take linearly independent basis functions $\varphi_i\in H_0^1(0,1)$ and write $u_h(x,t)=\sum_jU_j(t)\varphi_j(x)$. The <weak formulation> of the heat equation, tested against $\varphi_i$, is
$$
\int_0^1\partial_tu_h\,\varphi_i=-\int_0^1u_h'\varphi_i'.
$$
Thus the <semidiscrete finite element heat equation> is
$$
\boxed{M\dot U=KU,\qquad
M_{ij}=\int_0^1\varphi_i\varphi_j,\quad
K_{ij}=-\int_0^1\varphi_i'\varphi_j'.}
$$
The vector $U$ contains the basis coefficients. A natural initial vector $G$ is the $L^2$ projection of the initial data: $MG=b$, where $b_i=\int_0^1g\varphi_i$, and $U(0)=G$. Other consistent initial approximations can be specified, but exact representation of an arbitrary $g$ in the trial space is not implicit.
For a nonzero coefficient vector $a$, the function $v_h=\sum_ia_i\varphi_i$ is nonzero by basis independence. Hence $a^TMa=\|v_h\|_{L^2}^2>0$, proving that the <mass matrix> is symmetric positive definite. Also $a^TKa=-\|v_h'\|_{L^2}^2<0$: if the derivative vanished, $v_h$ would be a constant, and its zero endpoint traces would make it zero. Therefore \b[$K$ is symmetric negative definite], not merely semidefinite, for these homogeneous Dirichlet conditions.
The squared spatial norm is $U^TMU$. Differentiating and using the semidiscrete system gives the <energy identity>
$$
\frac d{dt}\|u_h(t)\|_{L^2}^2
=2U^TM\dot U=2U^TKU=-2\|u_h'(t)\|_{L^2}^2\leq0.
$$
Using the <trapezoidal rule> in time gives the <Crank-Nicolson method>
$$
\boxed{(M-\tfrac12\Delta t K)U^{k+1}
=(M+\tfrac12\Delta t K)U^k.}
$$
The left matrix is positive definite for every $\Delta t>0$, so every step exists uniquely. To prove stability, set $d=U^{k+1}-U^k$ and $s=U^{k+1}+U^k$. The update says $Md=(\Delta t/2)Ks$. Multiplying by $s^T$ gives
$$
(U^{k+1})^TMU^{k+1}-(U^k)^TMU^k
=\frac{\Delta t}{2}s^TKs\leq0.
$$
Consequently
$$
\boxed{\|u_h^{k+1}\|_{L^2}\leq\|u_h^k\|_{L^2}\quad
\text{for every }\Delta t>0.}
$$
The same inequality applied to the difference of two discrete solutions proves nonamplification of initial perturbations. This is unconditional stability in the mass-matrix norm, exactly the spatial $L^2$ norm of the finite element functions.
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