= Forward Euler limit for a consistent hat mass matrix
{title2=$0\leq\mu\leq\frac16\quad\text{uniformly over refinements}$}
For a uniform piecewise-linear Galerkin heat discretization, the consistent <mass matrix> is $M=(h/6)\operatorname{tridiag}(1,4,1)$ and the <stiffness matrix> is $K=h^{-1}\operatorname{tridiag}(-1,2,-1)$. The generalized <eigenvalues> are $6(1-\cos\theta)/[h^2(2+\cos\theta)]$. Applying <Forward Euler method> requires $k\lambda\leq2$, giving the displayed uniform bound for $\mu=k/h^2$. For $J$ intervals with Dirichlet endpoints the exact finite-grid upper bound is $(2-\cos(\pi/J))/[3(1+\cos(\pi/J))]$, which approaches $1/6$. Replacing the consistent <mass matrix> by a lumped one changes the <stability> restriction.
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