Forward Euler limit for a consistent hat mass matrix ($0\le \mu \le \frac{1}{6}\text{uniformly over refinements}$)

For a uniform piecewise-linear Galerkin heat discretization, the consistent mass matrix is $M=(h/6)\mathrm{tridiag}(1,4,1)$ and the stiffness matrix is $K=h^{-1}\mathrm{tridiag}(-1,2,-1)$. The generalized eigenvalues are $6(1-\cos \theta )/[h^2(2+\cos \theta )]$. Applying Forward Euler method requires $k\lambda \le 2$, 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.