Bounds on the eigenvalue range and the field of values of non-Hemitian and indefinite element matrices

In the early seventies, Fried formulated bounds on the spectrum of assembled Hermitian positive (semi-) definite finite element matrices using the extreme eigenvalues of the element matrices. In this paper we will generalise these results by presenting bounds on the field of values, the numerical radius and on the spectrum of general, possibly complex matrices, for both the standard and the generalised problem. The bounds are cheap to compute, involving operations with element matrices only. We illustrate our results with an example from acoustics involving a complex, non-Hermitian matrix. As an application, we show how our estimates can be used to derive an upper bound on the number of iterations needed to achieve a given residual reduction in the GMRES-algorithm for solving linear systems. Keywords: Eigenvalue bounds; Field of values; Numerical radius; Non-Hermitian finite element matrices; Iterative methods