Definition of the problem:
Throughout, will
denote a
bounded domain in
( or ).
In fact will
be assumed to be a polygon ( ) or
polyhedron ( ). We will be concerned with the
problem
of
finding an eigenvalue
and eigenfunction
satisfying
for all 
(1.1) 
where
and

(1.2) 
Here, the matrixvalued function
is required to be
uniformly positive definite,
i.e.
for all
with
and all 
(1.3) 
The methods which we describe below can be extended to piecewise smooth
coefficients
, but to reduce technical detail in the proofs we will assume in fact
that
is piecewise constant on and
that the jumps
in
are aligned with the meshes
(introduced below),
for all . The scalar function
is required to be
bounded above and below by
positive constants for all
, i.e.
for all

(1.4) 
Throughout the
paper, for any polygonal (polyhedral)
subdomain of
, and any
,
and
will denote the
standard norm and seminorm in the Sobolev space .
Also
denotes the inner product. We
also
define the energy norm
induced by the bilinear form :
for
all
which, by (1.3),
is equivalent to the
seminorm. (The equivalence constant depends on
the contrast
, but we are not
concerned with this dependence in the present paper.) We also introduce
the weighted norm:
and note the norm equivalence

(1.5) 
Rewriting the
eigenvalue problem (1.1)
in standard normalised form, we seek
such
that

(1.6) 
By the continuity
of and and
the coercivity of on
it is a standard
result that (1.6)
has a countable sequence of nondecreasing
positive eigenvalues
,
with corresponding eigenfunctions
.
We will need some
additional regularity for the
eigenfunctions , which will be
achieved by making the following regularity assumption for the elliptic
problem induced by :
Assumption 1.1
We assume that
there exists a constant
and
with the following property. For
, if
solves the problem
for
all
, then
.
Assumption 1.1
is satisfied with when
is constant (or
smooth) and is convex. In a range
of other practical cases
,
for example nonconvex, or
having a
discoutinuity
across an interior
interface. Under Assumption 1.1
it follows
that the eigenfunctions of
the problem (1.6)
satisfy
.
To approximate
problem (1.6)
we
use the continuous linear finite element method. Accordingly, let
denote
a family of
conforming triangular ( ) or
tetrahedral ( ) meshes on .
Each mesh consists of elements denoted
. We assume that for
each ,
is a refinement of
. For a typical
element of any mesh, its
diameter is denoted and
the diameter of its largest inscribed ball
is denoted . For each ,
let denote
the piecewise constant mesh function on ,
whose value on each element
is and
let
.
Throughout we
will assume that the family of meshes
is shape regular,
i.e.
there exists
a constant
such that
In the later sections of the paper the
will be produced by
an
adaptive
process which ensures shape regularity.
We let
denote the usual
finite dimensional subspace of
, consisting of all
continuous
piecewise linear functions with respect to the mesh
. Then the discrete
formulation of
problem (1.6)
is to seek the eigenpairs
such
that

(1.8) 
The problem (1.8)
has
positive eigenvalues (counted according to
multiplicity) which we denote in nondecreasing order as
.