Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 60 / 1

Put $h=1/(m+1)$ and impose homogeneous Dirichlet boundary conditions, as required by the stated continuum eigenvalues. Let $T=\mathrm{tridiag}(1,-2,1)$ be the unscaled one-dimensional Dirichlet discrete Laplacian. For $1\le k\le m$, the vectors $v_j^{(k)}=\sin (jk\pi h)$ satisfyThe identity follows by adding the two neighboring sines; the values at $j=0,m+1$ vanish. These $m$ mutually orthogonal vectors form the discrete sine transform basis. Consequently their tensor products give all $m^2$ two-dimensional modes, not merely a few candidate eigenvectors.

In a compatible grid ordering the axial operator is $A_5=h^{-2}(T\otimes I+I\otimes T)$. The nine-point operator isIndeed, the added Kronecker product supplies the diagonal neighbors with weight $1/6$, subtracts $1/3$ from each axial neighbor and adds $2/3$ to the central coefficient. Thus it has exactly the coefficients of the second discretization. On $v^{(k)}\otimes v^{(l)}$ the eigenvalues areandEquivalently, $h^2{\lambda }_{9,kl}=-10/3+4[\cos (k\pi h)+\cos (l\pi h)]/3+2\cos (k\pi h)\cos (l\pi h)/3$.

For fixed mode numbers $k,l$, expansion of the sines giveswhereas the mixed Kronecker product correction givesThus both eigenvalue approximations are second order for fixed modes. The leading nine-point error includes an additional positive term ${\pi }^4{h}^2{k}^2{l}^2/6$.

There is also an exact comparison, not just an asymptotic one. Since $\sin x<x$ for $0<x<\pi /2$, and both sine factors in the correction are positive,Both estimates lie on the less-negative side of the continuum Laplacian eigenvalue, so the five-point formula is the better eigenvalue estimate for every corresponding interior mode. The five-point versus nine-point Laplacian eigenvalue accuracy comparison does not contradict the useful isotropy or special Poisson accuracy properties of a nine-point stencil. Those properties concern a different error criterion. The fixed-mode expansion is not uniform for modes with $k$ or $l$ comparable to $h^{-1}$; wavelengths close to the mesh scale need not have small relative spectral error.