= Trapezoidal contractivity for a semidiscrete heat equation
{title2=$\|U^{k+1}\|_M\leq\|U^k\|_M$}
If $M$ is symmetric positive definite and $K$ symmetric negative semidefinite, the update $(M-\Delta tK/2)U^{k+1}=(M+\Delta tK/2)U^k$ is uniquely solvable for every positive step. Multiply its difference form by $(U^{k+1}+U^k)^T$ to obtain
$$
\|U^{k+1}\|_M^2-\|U^k\|_M^2
=\frac{\Delta t}2(U^{k+1}+U^k)^TK(U^{k+1}+U^k)\leq0.
$$
For finite element heat flow, the mass norm is precisely the spatial $L^2$ norm, giving unconditional stability of the <Crank-Nicolson method>.
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