Upper and lower bounds for eigenvalues by finite differences (Q2541541)

Explicit upper and lower bounds are obtained for the eigenvalues of the problems: \[ \Delta u + \lambda u = 0 \quad\text{in } R, \quad u=0 \quad\text{on }\partial R; \] \[ \Delta^2 v - \Omega v = 0 \quad\text{in } R, \quad v= \partial v/\partial n=0 \quad\text{on }\partial R; \] \[ \Delta^2 w - \Lambda\Delta w = 0 \quad\text{in } R, \quad w= \partial w/\partial n=0 \quad\text{on }\partial R; \] for \(R\) a bounded region in \(n\)-space. The bounds are in terms of the corresponding eigenvalues of the appropriate finite-difference analogues, and are obtained by variational methods 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.