Preconditioning by inverting the Laplacian: an analysis of the eigenvalues

We study the eigenvalues of the operator generated by using the inverse of the Laplacian as a preconditioner for self-adjoint second-order elliptic partial differential equations with smooth coefficients. It is well-known that the spectral condition number of the preconditioned operator can be bounded by $\frac{\max k(x)}{\min k(x)}$, where k is the uniformly positive coefficient of the second-order elliptic equation. The purpose of this paper is to study the spectrum of the preconditioned operator. We will show that there is a strong relation between the spectrum of this operator and the range of the coefficient function. In the continuous case, we prove, both for mappings defined on Sobolev spaces and in terms of generalized functions, that the spectrum of the preconditioned operator contains the range of the coefficient function k. In the discrete case, we indicate by numerical examples that the entire discrete spectrum is approximately given by values of k.