= Centered discrete advection is skew-adjoint
{title2=$\operatorname{Re}\langle u,D_0u\rangle_h=0$}
With zero endpoints, the centered first-difference matrix has opposite off-diagonal entries and is a <skew-adjoint operator> in the mesh-weighted Euclidean inner product. Adding a real multiple of it to the negative symmetric second-difference matrix does not change the real energy form. The semidiscrete convection-diffusion solution therefore obeys a mesh-independent <energy estimate>, even when its drift is large compared with its diffusion coefficient.
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