The main result of
this paper is Theorem 3.2
below which proves convergence of the adaptive
method and also demonstrates the decay of oscillations of the sequence
of approximate eigenfunctions. Before proving this result we need a
final lemma.

Theorem 3.2Provided the
initial mesh
is chosen so that
is small enough, there exists a
constant
such that the recursive application of Algorithm 1
yields a convergent sequence of
approximate eigenvalues and eigenvectors, with the property:

(3.1)

and

(3.2)

where and are
positive constants.

Remark
3.3The initial mesh
convergence threshold and the
constants and may
depend on ,
and .