Upper and lower bounds
for the eigenvalues of elliptic partial differential equations associated with
fixed membranes and clamped plates are given in terms of corresponding
eigenvalues of their finite difference analogues. The upper bounds are found by
interpolating piecewise polynomials through the solutions to the difference
equations and substituting into the variational principle associated with the
differential equations. The lower bounds are found by averaging the solutions
to the differential equations and substituting into the discrete variational
principle.
