 Solution

Test with $v_h=u_h^{m+1}$ and apply $2(a-b,a)=\Vert a{\Vert }^2-\Vert b{\Vert }^2+\Vert a-b{\Vert }^2$. The exact energy identity isThe printed identity omits the last reaction contribution while explicitly describing its last term as the $H^1$ seminorm. It requires the full $H^1$ norm instead of that seminorm. With the printed seminorm its left side equals $-\Vert u_h^{m+1}{\Vert }_2^2$, rather than zero. The source reaction term forces this correction.

All terms other than the difference of successive squared norms are nonnegative. ConsequentlyThe second inequality uses contractivity of the initial orthogonal projection. No restriction relates $\Delta t$ to $h$, giving unconditional stability. The retained reaction term strengthens dissipation.