= Skew symmetry of Galerkin advection
{title2=$\int_\Omega v\,\mathbf b\cdot\nabla v=0$}
For a divergence-free drift $\mathbf b$ and a <function> $v$ with zero boundary <Sobolev trace>, <integration by parts> gives $\int_\Omega v\,\mathbf b\cdot\nabla v=\tfrac12\int_{\partial\Omega}v^2\mathbf b\cdot\mathbf n-\tfrac12\int_\Omega(\nabla\cdot\mathbf b)v^2=0$. Consequently the advection matrix in a conforming <Galerkin method> has zero <quadratic form>; its symmetric part vanishes. Adding a diffusion <stiffness matrix> gives a nonpositive spatial generator and a nonincreasing <L2 norm> in the semidiscrete <advection-diffusion equation>. The same identity holds with a suitable zero normal-flux boundary term instead of zero trace.
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