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 Incoming links: High-frequency eigenvalue error of the discrete Laplacian

Past exam of the numerical solutions of partial differential equations course of the University of Oxford / 2023 / 2 / d / Solution 2026-10-25

For every admissible

m, n

, the trigonometric inequality

sin z \leq z

for

z \geq 0

gives

\frac{4}{h ^{2}} sin^{2} \frac{mπh}{2} \leq m^{2} π^{2}, \frac{4}{h ^{2}} sin^{2} \frac{nπh}{2} \leq n^{2} π^{2} .

Thus

Λ_{m, n} \leq λ_{m, n}

, in particular for

m = n = N - 1

and every

N \geq 3

.

For these highest-index eigenvalues, use

h = 1/ N

and

sin ((N - 1) π / (2 N)) = cos (π / (2 N))

to obtain

λ_{N - 1, N - 1} - Λ_{N - 1, N - 1} = 2 π^{2} (N - 1)^{2} - 8 N^{2} cos^{2} \frac{π}{2 N} .

Divide by

N^{2}

and take the limit:

N \to \infty lim \frac{λ _{N - 1, N - 1} - Λ _{N - 1, N - 1}}{N ^{2}} = 2 π^{2} - 8 > 0.

For example, eventually the normalized difference exceeds

(2 π^{2} - 8) /2

, so the unnormalized difference is at least

(π^{2} - 4) N^{2}

and tends to

+ \infty

. This does not contradict (c): there the indices stay fixed as the grid is refined; here the eigenfunctions oscillate increasingly quickly, with wavelength comparable to the mesh spacing. It illustrates the high-frequency eigenvalue error of the discrete Laplacian.

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