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

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 $d(U^{T}MU)/dt=-2U^{T}KU\le 0$.

The sine vectors $v_j^{(m)}=\sin (j{\theta }_m)$, ${\theta }_m=m\pi /J$, simultaneously diagonalize these two symmetric tridiagonal matrices. Their eigenvalues areThus the semidiscrete decay rates are ${\lambda }_m=6(1-\cos {\theta }_m)/[h^2(2+\cos {\theta }_m)]$. The forward Euler method multiplies each mode by $1-k{\lambda }_m$. Stability requires $0\le k{\lambda }_m\le 2$ for every mode.

For a fixed finite Dirichlet grid the largest mode is ${\theta }_{J-1}=\pi -\pi /J$. Consequently the exact finite-grid bound isAs the mesh is refined, the upper bound decreases to $1/6$. Hence the mesh-independent answer is $\boxed{0\le \mu \le 1/6}$. This is the forward Euler limit for a consistent hat mass matrix. The frequently seen $1/2$ diffusion bound belongs to the standard second-difference scheme or a lumped mass matrix, and is not the bound for the consistent Galerkin equations derived here.