= Energy stability for variable-coefficient reaction diffusion
{title2=$\|U(t)\|_h\leq e^{a_+t}\|U(0)\|_h$}
For a zero-boundary centered semidiscretization $U'=D_hU+V_hU$, where $D_h$ is the scaled <Dirichlet discrete Laplacian> and $V_h=\operatorname{diag}(a_j)$ with $a_j\leq a_+$, <summation by parts> gives $(U,D_hU)_h\leq0$. Hence $\tfrac12(d/dt)\|U\|_h^2\leq a_+\|U\|_h^2$. The <Gronwall inequality> yields $\|U(t)\|_h\leq e^{a_+t}\|U(0)\|_h$, a mesh-independent <stability> bound on every fixed finite interval. Positive reaction coefficients can permit physical growth without violating this form of <stability>.
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