= Solution
Use $J$ equal intervals, $h=1/J$, and the interior <piecewise-linear hat functions> $\phi_j$, which equal one at node $j$, vanish at all other nodes and vanish outside the two adjacent intervals. Write $u_h(x,t)=\sum_{j=1}^{J-1}U_j(t)\phi_j(x)$. Testing with $\phi_i$ and integrating the second <derivative> by parts gives
$$
\sum_j\left(\int_0^1\phi_i\phi_jdx\right)\dot U_j
=-\sum_j\left(\int_0^1\phi_i'\phi_j'dx\right)U_j.
$$
The integrals are local: $\int\phi_i^2=2h/3$, $\int\phi_i\phi_{i+1}=h/6$, $\int(\phi_i')^2=2/h$ and $\int\phi_i'\phi_{i+1}'=-1/h$. Nonadjacent functions do not overlap. Thus the consistent <mass matrix> and <stiffness matrix> are
$$
M=\frac h6\operatorname{tridiag}(1,4,1),\qquad
K=\frac1h\operatorname{tridiag}(-1,2,-1),\qquad \boxed{M\dot U+KU=0.}
$$
Equivalently the explicit nodal equations are
$$
\boxed{\dot U_{j-1}+4\dot U_j+\dot U_{j+1}
=\frac6{h^2}(U_{j-1}-2U_j+U_{j+1}),\quad U_0=U_J=0.}
$$
For merely L2 initial data a natural choice is the Galerkin projection $MU(0)=((v,\phi_i))_i$; nodal interpolation is also possible for suitably smooth data. The relation $U^TMU=\|u_h\|_{L^2}^2$ identifies the natural finite-element energy. Differentiating gives $d(U^TMU)/dt=-2U^TKU\leq0$.
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