= Reaction contribution in backward Euler energy dissipation
{c}
{title2=$\frac{\|u^{m+1}\|_2^2-\|u^m\|_2^2+\|u^{m+1}-u^m\|_2^2}{2\Delta t}+\|u^{m+1}\|_{H^1}^2=0$}
For $u_t-u_{xx}+u=0$ with homogeneous mixed Dirichlet and <Neumann boundary conditions>, testing the <Backward Euler method> <finite element> equation with the new solution gives the displayed identity. Both the squared <derivative> <norm> and the squared function <norm> are required in the dissipation, because diffusion and reaction contribute separately. Replacing the full $H^1$ <norm> by its seminorm omits the reaction term. Positivity gives unconditional decay of the $L^2$ <norm>.
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