Bounds and error control for eigenvalues

Abstract

Control and estimation of errors are important but difficult aspects of any analysis from which the numerical results are necessarily approximate. The order of difficulty is greater for local or distributed quantities like stresses and displacements than for global Of integrated parameter~ like eigenvalues and stiffnesses. To really bound a desired quantity between a pair of dose upper and lower bounds one should obtain either an oscillatory but clear convergence or, preferably, two rapidly converging sequences one from above and the other from below. Application of the two complementary variational principles of energy and complimentary energy, when both are possible to apply, do yield upper and lower bound approximations. But these or other alternate methods are generally expensive. On the other hand it would be advantageous if one basic procedure could be perturbed in a simple manner to provide both lower and upper bounds and to refine the solution and control the errors without undue effort. This paper discusses this concept and presents three powerful methods to closely bound any desired parameter in a problem. These are particularly valuable for eigenvalue problems.