A new mixed clustered element-by-element (CEBE)/cluster companion (CC)
preconditioning method for finite element computations is introduced. 
In the CEBE preconditioning, the elements are merged into clusters 
of elements, and the preconditioners are defined as series products 
of cluster level matrices. The CC preconditioning method, which 
is also introduced in this paper, shares a common philosophy 
with the multi-grid methods. The CC preconditioners are based 
on companion meshes associated with different levels of clustering. 
For each level of clustering, we construct a CEBE preconditioner 
and an associated CC preconditioner. Because these two preconditioners 
in a sense complement each other, when they are used in a mixed 
way, they can be expected to give better performance. In fact, 
our numerical tests, for two- and three-dimensional problems 
governed by the Poisson equation, demonstrate that the mixed 
CEBE/CC preconditioning results in convergence rates which are, 
in most cases, significantly better than the convergence rates 
obtained with the best of the CEBE and CC preconditioning methods.