Convergence1
Convergence for Eigenvalue Problems with Mesh Adaptivity
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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 matrix-valued 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 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 non-decreasing 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 non-convex, 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 
for all 
and all 
(1.7) 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 non-decreasing order as 
.