  A method of estimating all eigenvalues of a preconditioned discretized scalar
diffusion operator with Dirichlet boundary conditions has been recently
introduced in T. Gergelits, K.A. Mardal, B.F. Nielsen, Z. Strako\v{s}:
Laplacian preconditioning of elliptic PDEs: Localization of the eigenvalues of
the discretized operator, SIAM Journal on Numerical Analysis 57(3) (2019),
1369-1394. Motivated by this paper, we offer a slightly different approach that
extends the previous results in some directions. Namely, we provide bounds on
all increasingly ordered eigenvalues of a general diffusion or elasticity
operator with tensor data, discretized with the conforming finite element
method, preconditioned by the inverse of a matrix of the same operator with
different data. Our results hold for mixed Dirichlet and Robin or periodic
boundary conditions applied to the original and preconditioning problems. The
bounds are two-sided, guaranteed, easily accessible, and depend solely on the
material data.
