Reaction contribution in backward Euler energy dissipation ($\frac{\Vert u^{m+1}{\Vert }_2^2-\Vert u^m{\Vert }_2^2+\Vert u^{m+1}-u^m{\Vert }_2^2}{2\Delta t}+\Vert u^{m+1}{\Vert }_{H^1}^2=0$)

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.