A stretched rubber membrane (a drumhead) is clamped at its square edge of side and…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics · Numerical and computational mathematics

Error, Chance and the Machine, Ch Partial differential equations: stability, convergence and solvers

Problem

A stretched rubber membrane (a drumhead) is clamped at its square edge of side

10\,\mathrm{cm}

and pushed down by a uniform pressure, so its small transverse deflection

u(x,y)

satisfies the Poisson equation

-\nabla^2 u = f

with

u = 0

on the boundary. Discretised on an

N\times N

grid of interior points with the five-point stencil, this gives the same symmetric positive-definite linear system as the heated plate of this chapter. On that plate at

N = 100

, starting from zero, conjugate gradients took

159

iterations to reduce the residual by a factor of

10^{6}

, and optimal SOR took

269

sweeps to reduce the error by the same factor. The membrane is now to be solved on a finer grid,

N = 400

. ((a)) From the Jacobi spectral radius

\rho_{\rm J} = \cos\bigl(\pi/(N+1)\bigr)

, find

\rho_{\rm J}

, the best SOR parameter

\omega

and

\rho_{\rm SOR} = \omega - 1

N = 400

. Hence estimate the Jacobi and optimal-SOR sweep counts for a reduction by

10^{6}

from

\ln(10^{6})/(-\ln\rho)

. ((b)) Using how each count scales with

N

, predict from the measured counts at

N = 100

the number of conjugate-gradient iterations and optimal-SOR sweeps that a run at

N = 400

would take. Say why the SOR prediction is larger than the estimate of part (a). ((c)) State the one piece of information optimal SOR needs to reach this rate that conjugate gradients do not.

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