LOWER BOUNDS FOR HIGHER EIGENVALUES OF SECOND ORDER OPERATORS BY FINITE DIFFERENCE METHODS
Abstract
Lower bounds for all the eigenvalues or an arbitrary second order self-adjoint elliptic differential operator on a bounded domain R with zero boundary conditions are given in terms of the eigenvalues of an associated finite difference problem. When r is sufficien ly smooth, the lower bounds converge to the eigenvalues themselves as the mesh size approaches zero. A certain class of selfl equations containing no mixed derivatives is also treated. Upper bounds for the eigenvalues of a differential operator can always be found by the Rayleigh-Ritz method. That is, one puts piecewise differentiable functions vanishing on the boundary into the Poincare inequality. It was pointed out by Courant that in the case of second order operators one can reduce the problem of upper bounds to a finite difference eigenvalue adjoint systems of elliptic differential equations containing no mixed derivatives is also treated. Upper bounds for the eigenvalues of a differential operator can always be found by the Rayleigh-Ritz method. That is, one puts piecewise differentiable functions vanishing on the boundary into the Poincare inequality. It was pointed out by Courant that in the case of second order operators one can reduce the problem of upper bounds to a finite difference eigenvalue problem by using piecewise linear functions. (Author)
Document Details
- Document Type
- Technical Report
- Publication Date
- Aug 26, 1957
- Accession Number
- AD0286398
Entities
People
- H.f. Weinberger
Organizations
- University of Wisconsin–Madison