Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 69 / 5 / a / Solution

Use $J$ equal intervals, $h=1/J$, and the interior piecewise-linear hat functions ${\varphi }_j$, which equal one at node $j$, vanish at all other nodes and vanish outside the two adjacent intervals. Write $u_h(x,t)={\sum }_{j=1}^{J-1}{U}_j(t){\varphi }_j(x)$. Testing with ${\varphi }_i$ and integrating the second derivative by parts givesThe integrals are local: $\int {\varphi }_i^2=2h/3$, $\int {\varphi }_i{\varphi }_{i+1}=h/6$, $\int ({\varphi }_i^{\prime }{)}^2=2/h$ and $\int {\varphi }_i^{\prime }{\varphi }_{i+1}^{\prime }=-1/h$. Nonadjacent functions do not overlap. Thus the consistent mass matrix and stiffness matrix areEquivalently the explicit nodal equations areFor merely L2 initial data a natural choice is the Galerkin projection $MU(0)=((v,{\varphi }_i){)}_i$; nodal interpolation is also possible for suitably smooth data. The relation $U^{T}MU=\Vert u_h{\Vert }_{L^2}^2$ identifies the natural finite-element energy. Differentiating gives $d(U^{T}MU)/dt=-2U^{T}KU\le 0$.