= Semidiscrete finite element heat equation
{title2=$M\dot c+Sc=F$}
The spatial <Galerkin method> for the <heat equation> gives a <mass matrix> $M$ and a diffusion <stiffness matrix> $S$. With zero forcing, $\frac{d}{dt}(c^TMc)=-2c^TSc\leq0$. The <quadratic form> $c^TMc$ is the squared <L2 norm> of the represented finite-element function, proving an energy contraction. For nonzero forcing, the corresponding identity has the additional term $2c^TF$, which can be estimated using the <Cauchy-Schwarz inequality>.
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